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Визуализация чисто геометрического вывода формулы определителя третьего порядка

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В этой статье будет исключительно визуализация вывода формул чисто геометрически. Соответственно, она для тех, кто уже и так хорошо знаком с темой и уже знает, что такое определитель и с чем его едят, хотя может быть использована при изучении этой темы, когда уже изучены базовые понятия и хочется ознакомиться с геометрическим выводом, а также просто посмотреть красивые картинки.

Геометрический вывод в 2D — вычисление (ориентированной) площади серого параллелограмма на картинке:

$\det(\vec{a}; \vec{b}) = \det((a_1; a_2); (b_1; b_2)) = (a_1+b_1)(a_2+b_2) - a_1a_2-b_1b_2-2b_1a_2= \\ =a_1b_2-b_1a_2$

Данную картинку многие уже знают, некоторые даже сами её рисовали и выводили формулу, когда изучали определители.

Переход в 3D

В трёхмерном пространстве картина выглядит куда неприятнее с вычислительной точки зрения. Если в 2D нужно было считать площадь 6 объектов и некоторые повторяются, то в 3D нужно считать объёмы аж 24 объектов, хотя некоторые повторяются, но менее монотонным и долгим процесс от этого не становится, поэтому вывод формулы в 3D оставим в качестве упражнения для читателя, а полностью на бумаге и без 3D движков вроде Desmos3D — это, наверняка, был сущий кошмар, требующий хорошего пространственного мышления, если вообще кто‑то делал так когда‑либо. Соответственно, в этой оболочке из фигур находится параллелепипед, построенный на трёх векторах (a; b; c), объёму которого по модулю равен определитель 3×3 матрицы с этими векторами в качестве столбцов или строк.

$|\det(\vec{a}, \vec{b}, \vec{c}) | = V_{пар.}$

Ссылки на Desmos и разметка:

2D, 3D

O=\left(0,0\right)
Folder: Vectors a and b
A=\left(6.47,3.09\right)
B=\left(2,5\right)
\operatorname{polygon}\left(O,A\right)
\operatorname{polygon}\left(O,B\right)
% ↓det(a; b)
A.xB.y-A.yB.x
% Shapes:
\operatorname{polygon}\left(A,A+B\right)
\operatorname{polygon}\left(B,A+B\right)
\operatorname{polygon}\left(O,A,A+B,B\right)
\operatorname{polygon}\left(O,\left(0,B.y\right),B\right)
\operatorname{polygon}\left(A,A+B,A+\left(B.x,0\right)\right)
\operatorname{polygon}\left(B,A+B,B+\left(0,A.y\right)\right)
\operatorname{polygon}\left(O,A,\left(A.x,0\right)\right)
\operatorname{polygon}\left(\left(0,B.y\right),B,B+\left(0,A.y\right),\left(0,A.y+B.y\right)\right)
\operatorname{polygon}\left(\left(A.x,0\right),A,A+\left(B.x,0\right),\left(A.x+B.x,0\right)\right)

Folder: BS [not related to the topic]
\operatorname{polygon}\left(A,A-A\cdot0.05\cos\theta_{0}+\left(A.y,-A.x\right)\cdot0.05\sin\theta_{0},A-A\cdot0.05\cos\theta_{0}-\left(A.y,-A.x\right)\cdot0.05\sin\theta_{0}\right)
\operatorname{polygon}\left(B,B-B\cdot0.05\cos\theta_{0}-\left(B.y,-B.x\right)\cdot0.05\sin\theta_{0},B-B\cdot0.05\cos\theta_{0}+\left(B.y,-B.x\right)\cdot0.05\sin\theta_{0}\right)
\theta_{0}=\frac{\pi}{20}

O=\left(0,0,0\right)
% ↑ Start of coordinate system
Folder: Points (actual vectors a, b, c)
A=\left(a_{1},a_{2},a_{3}\right)
B=\left(b_{1},b_{2},b_{3}\right)
C=\left(c_{1},c_{2},c_{3}\right)
Folder: Vector A
a_{1}=8
a_{2}=3.2
a_{3}=1.27
Folder: Vector B
b_{1}=2.1
b_{2}=8
b_{3}=1.67
Folder: Vector C
c_{1}=2.7
c_{2}=2.05
c_{3}=9.3
% 2D determinant on Oxy:
Folder: A and B projections onto Oxy plane
a_{0}=\left(a_{1},a_{2},0\right)
b_{0}=\left(b_{1},b_{2},0\right)
\operatorname{vector}\left(O,a_{0}\right)
\operatorname{vector}\left(O,b_{0}\right)
\operatorname{vector}\left(a_{0},a_{0}+b_{0}\right)
\operatorname{vector}\left(b_{0},a_{0}+b_{0}\right)
Folder: 1
\operatorname{triangle}\left(O,a_{0},\left(a_{1},0,0\right)\right)
\operatorname{triangle}\left(b_{0},a_{0}+b_{0},b_{0}+\left(0,a_{2},0\right)\right)
Folder: 2
\operatorname{triangle}\left(O,b_{0},\left(0,b_{2},0\right)\right)
\operatorname{triangle}\left(a_{0},a_{0}+b_{0},a_{0}+\left(b_{1},0,0\right)\right)
Folder: 3
\operatorname{triangle}\left(\left(a_{1}+b_{1},0,0\right),\left(a_{1},0,0\right),a_{0}+\left(b_{1},0,0\right)\right)
\operatorname{triangle}\left(a_{0},a_{0}+\left(b_{1},0,0\right),\left(a_{1},0,0\right)\right)
\operatorname{triangle}\left(b_{0},\left(0,b_{2}+a_{2},0\right),b_{0}+\left(0,a_{2},0\right)\right)
\operatorname{triangle}\left(b_{0},\left(0,b_{2}+a_{2},0\right),\left(0,b_{2},0\right)\right)
Folder: Parallelogram
\operatorname{triangle}\left(O,b_{0},b_{0}+a_{0}\right)
\operatorname{triangle}\left(O,a_{0},a_{0}+b_{0}\right)
% ===3D determinant in the space===
Folder: Main vectors a, b, c (drawings of their arrows in 3D)
a\ =\operatorname{vector}\left(O,\ A\right)
b=\operatorname{vector}\left(O,\ B\right)
c=\operatorname{vector}\left(O,\ C\right)
Folder: Other „vectors“ (actually same vectors but different arrows in space)
\operatorname{vector}\left(B,A+B\right)
\operatorname{vector}\left(A,A+B\right)
\operatorname{vector}\left(C,A+C\right)
\operatorname{vector}\left(C,\ B+C\right)
\operatorname{vector}\left(A,\ A+C\right)
\operatorname{vector}\left(B,B+C\right)
\operatorname{vector}\left(A+B,\ A+B+C\right)
\operatorname{vector}\left(C+A,\ A+B+C\right)
\operatorname{vector}\left(C+B,\ A+B+C\right)
Folder: Determinant (more precisely only a parallelepiped which volume equal to absolute value of det(a; b; c))
\operatorname{triangle}\left(O,C,A\right)
\operatorname{triangle}\left(A,C,A+C\right)
\operatorname{triangle}\left(O,B,A\right)
\operatorname{triangle}\left(B,A,A+B\right)
\operatorname{triangle}\left(A+C,A+B,A\right)
\operatorname{triangle}\left(A+B+C,A+C,A+B\right)
\operatorname{triangle}\left(A+B+C,B+C,A+B\right)
\operatorname{triangle}\left(A+B,B,B+C\right)
\operatorname{triangle}\left(O,C,B\right)
\operatorname{triangle}\left(B,B+C,C\right)
\operatorname{triangle}\left(C,A+C,A+B+C\right)
\operatorname{triangle}\left(C,A+B+C,B+C\right)
Folder: Planes
x=A.x+B.x+C.x
y=A.y+B.y+C.y
z=A.z+B.z+C.z
% ↓Prisms:
% Slider of parameter m for moving some objects to connect them by subtracting or adding some vector or combination of vectors scaled by m
m=0
Folder: 1.1
\operatorname{triangle}\left(O,A,\left(A.x,\ A.y,0\right)\right)
\operatorname{triangle}\left(O,B,\left(B.x,\ B.y,0\right)\right)
\operatorname{triangle}\left(O,A,B\right)
\operatorname{triangle}\left(A,B,A+B\right)
\operatorname{triangle}\left(\left(A.x,\ A.y,0\right),A+B,\ A\right)
\operatorname{triangle}\left(\left(A.x,\ A.y,0\right),A+B,\ \left(\left(A+B\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(A+B,\left(B.x,\ B.y,0\right),\left(\left(A+B\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(A+B,\left(B.x,\ B.y,0\right),B\right)
\operatorname{triangle}\left(O,\left(A.x+B.x,\ A.y+B.y,\ 0\right),\left(A.x,A.y,0\right)\right)
\operatorname{triangle}\left(O,\left(A.x+B.x,\ A.y+B.y,\ 0\right),\left(B.x,B.y,0\right)\right)
Folder: 1.2
\operatorname{triangle}\left(A+C-mC,A+C+\left(0,0,B.z\right)-mC,A+B+C-mC\right)
\operatorname{triangle}\left(A+C-mC,A+C+\left(0,0,B.z\right)-mC,C-mC\right)
\operatorname{triangle}\left(C-mC,C+\left(0,0,A.z+B.z\right)-mC,A+C+\left(0,0,B.z\right)-mC\right)
\operatorname{triangle}\left(C-mC,C+\left(0,0,A.z+B.z\right)-mC,B+C-mC\right)
\operatorname{triangle}\left(C+\left(0,0,A.z+B.z\right)-mC,B+C-mC,B+C+\left(0,0,A.z\right)-mC\right)
\operatorname{triangle}\left(C-mC,A+C-mC,A+B+C-mC\right)
\operatorname{triangle}\left(C-mC,A+B+C-mC,B+C-mC\right)
\operatorname{triangle}\left(B+C+\left(0,0,A.z\right)-mC,A+B+C-mC,B+C-mC\right)
\operatorname{triangle}\left(A+B+C-mC,\ C+\left(0,0,A.z+B.z\right)-mC,B+C+\left(0,0,A.z\right)-mC\right)
\operatorname{triangle}\left(C+\left(0,0,B.z+A.z\right)-mC,A+C+\left(0,0,B.z\right)-mC,A+B+C-mC\right)
Folder: 2.1
\operatorname{triangle}\left(O,C,\left(0,C.y,C.z\right)\right)
\operatorname{triangle}\left(O,B,\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(O,B+C,B\right)
\operatorname{triangle}\left(O,B+C,C\right)
\operatorname{triangle}\left(\left(0,C.y,C.z\right),B+C,C\right)
\operatorname{triangle}\left(\left(0,C.y,C.z\right),B+C,\left(0,\left(B+C\right).y,\left(B+C\right).z\right)\right)
\operatorname{triangle}\left(B+C,\left(0,B.y,B.z\right),B\right)
\operatorname{triangle}\left(\left(0,\left(B+C\right).y,\left(B+C\right).z\right),B+C,\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(\left(0,\left(B+C\right).y,\left(B+C\right).z\right),O,\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(\left(0,\left(B+C\right).y,\left(B+C\right).z\right),O,\left(0,C.y,C.z\right)\right)
Folder: 2.2
\operatorname{triangle}\left(A-mA,A+\left(B.x+C.x,0,0\right)-mA,A+B-mA\right)
\operatorname{triangle}\left(A+B+\left(C.x,0,0\right)-mA,A+\left(B.x+C.x,0,0\right)-mA,A+B-mA\right)
\operatorname{triangle}\left(A+B+C-mA,A+B-mA,A+B+\left(C.x,0,0\right)-mA\right)
\operatorname{triangle}\left(A-mA,A+\left(B.x+C.x,0,0\right)-mA,A+C+\left(B.x,0,0\right)-mA\right)
\operatorname{triangle}\left(A+C-mA,A-mA,A+C+\left(B.x,0,0\right)-mA\right)
\operatorname{triangle}\left(A+C-mA,\ A+C+\left(B.x,0,0\right)-mA,\ A+B+C-mA\right)
\operatorname{triangle}\left(A+B+C-mA,\ A-mA,\ A+B-mA\right)
\operatorname{triangle}\left(A-mA,\ A+B+C-mA,\ A+C-mA\right)
\operatorname{triangle}\left(A+\left(C.x+B.x,0,0\right)-mA,A+B+C-mA,A+B+\left(C.x,\ 0,0\right)-mA\right)
\operatorname{triangle}\left(A+\left(C.x+B.x,0,0\right)-mA,A+C+\left(B.x,0,0\right)-mA,A+B+C-mA\right)
Folder: 3.1
\operatorname{triangle}\left(O,A,\left(A.x,0,A.z\right)\right)
\operatorname{triangle}\left(O,C,\left(C.x,0,C.z\right)\right)
\operatorname{triangle}\left(\left(C.x,0,C.z\right),C,A+C\right)
\operatorname{triangle}\left(\left(C.x,0,C.z\right),C+A,\left(A.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x+C.x,0,A.z+C.z\right),A+C,A\right)
\operatorname{triangle}\left(A,\left(A.x,0,A.z\right),\left(A.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(O,A+C,C\right)
\operatorname{triangle}\left(O,A+C,A\right)
\operatorname{triangle}\left(O,\left(C.x,0,C.z\right),\left(A.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(O,\left(A.x+C.x,0,A.z+C.z\right),\left(A.x,0,A.z\right)\right)
Folder: 3.2
\operatorname{triangle}\left(B-mB,B+C-mB,A+B+C-mB\right)
\operatorname{triangle}\left(B-mB,A+B+C-mB,A+B-mB\right)
\operatorname{triangle}\left(B-mB,\ B+C-mB,\left(B.x,\ B.y+A.y+C.y,B.z\right)-mB\right)
\operatorname{triangle}\left(\left(B.x,\ B.y+A.y+C.y,B.z\right)-mB,B+C-mB,B+C+\left(0,A.y,0\right)-mB\right)
\operatorname{triangle}\left(B+C+\left(0,A.y,0\right)-mB,A+B+C-mB,B+C-mB\right)
\operatorname{triangle}\left(A+B-mB,A+B+\left(0,C.y,0\right)-mB,A+B+C-mB\right)
\operatorname{triangle}\left(A+B-mB,A+B+\left(0,C.y,0\right)-mB,B-mB\right)
\operatorname{triangle}\left(A+B+\left(0,C.y,0\right)-mB,B-mB,\left(B.x,\ B.y+A.y+C.y,B.z\right)-mB\right)
\operatorname{triangle}\left(A+B+C-mB,\left(B.x,\ B.y+A.y+C.y,B.z\right)-mB,B+C+\left(0,A.y,0\right)-mB\right)
\operatorname{triangle}\left(A+B+C-mB,\left(B.x,\ B.y+A.y+C.y,B.z\right)-mB,B+A+\left(0,C.y,0\right)-mB\right)
% ↓Other parts:
Folder: 1.1
\operatorname{triangle}\left(C+\left(0,0,A.z+B.z\right),B+C,B+C+\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(C,C+\left(0,0,A.z+B.z\right),B+C\right)
\operatorname{triangle}\left(C,\left(0,C.y,C.z\right),\left(0,C.y,C.z+A.z+B.z\right)\right)
\operatorname{triangle}\left(C,C+\left(0,0,A.z+B.z\right),\left(0,C.y,C.z+A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+A.z+B.z\right),C+\left(0,0,A.z+B.z\right),B+C+\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+A.z+B.z\right),B+C+\left(0,0,A.z\right)-\left(C.x+B.x,0,0\right),B+C+\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z\right),B+C,\left(0,\left(B+C\right).y,\left(B+C\right).z\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z\right),B+C,\left(0,\left(B+C\right).y,\left(B+C\right).z\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z\right),B+C,C\right)
\operatorname{triangle}\left(\left(0,C.y,C.z\right),\left(0,C.y,C.z+A.z+B.z\right),\left(0,B.y+C.y,B.z+C.z\right)\right)
\operatorname{triangle}\left(B+C+\left(0,0,A.z\right)-\left(B.x+C.x,0,0\right),\left(0,C.y,C.z+A.z+B.z\right),\left(0,B.y+C.y,B.z+C.z\right)\right)
\operatorname{triangle}\left(B+C,B+C+\left(0,0,A.z\right),B+C+\left(0,0,A.z\right)-\left(B.x+C.x,0,0\right)\right)
\operatorname{triangle}\left(B+C,\left(0,B.y+C.y,B.z+C.z\right),B+C+\left(0,0,A.z\right)-\left(B.x+C.x,0,0\right)\right)
Folder: 1.2 MOVABLE
\operatorname{triangle}\left(A+B+\left(C.x,0,0\right)+v-mB,A+\left(B.x+C.x,0,0\right)+v+mB,A+B+v-mB\right)
\operatorname{triangle}\left(A+v+mB,A+\left(B.x+C.x,0,0\right)+v+mB,A+B+v-mB\right)
\operatorname{triangle}\left(\left(A.x,\ A.y,0\right)+v+m\left(0,0,A.z\right)+m\left(C.x,0,0\right),A+B+v+m\left(0,0,A.z\right)+m\left(C.x+B.x,0,0\right),\ A+v+m\left(0,0,A.z+B.z\right)+m\left(C.x,0,0\right)\right)
\operatorname{triangle}\left(\left(A.x,\ A.y,0\right)+v+m\left(C.x,0,A.z\right),A+B+v+m\left(C.x+B.x,0,A.z\right),\left(\left(A+B\right).x,\left(A+B\right).y,0\right)+v+m\left(B.x+C.x,0,B.z+A.z\right)\right)
\operatorname{triangle}\left(A+B+\left(C.x,0,0\right)+v-mB+m\left(0,0,A.z+B.z\right),A+B+v-mB+m\left(0,0,A.z+B.z\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)+v-mB+m\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(\left(A+B\right).x,\left(A+B\right).y,0\right)+v-mB+m\left(0,0,A.z+B.z\right),A+B+v-mB+m\left(0,0,A.z+B.z\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)+v-mB+m\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)+v+mB+m\left(0,0,A.z\right),A+v+mB+m\left(0,0,A.z\right),A+\left(B.x+C.x,0,0\right)+v+mB+m\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)+v+mB+m\left(0,0,A.z\right),A+\left(B.x+C.x,0,0\right)+v+mB+m\left(0,0,A.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)+v+mB+m\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)+v+m\left(B.x,B.y,0\right)+m\left(0,0,B.z+2A.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)+v+m\left(B.x,B.y,0\right)+m\left(0,0,B.z+2A.z\right),\left(\left(A+B\right).x,\left(A+B\right).y,0\right)+v-m\left(B.x,B.y,0\right)+m\left(0,0,B.z+2A.z\right)\right)
\operatorname{triangle}\left(\left(\left(A+B\right).x,\left(A+B\right).y,0\right)+v+m\left(0,0,2A.z+B.z\right)-m\left(B.x,B.y,0\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)+v+m\left(0,0,2A.z+B.z\right)-m\left(B.x,B.y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)+v+m\left(B.x,B.y,0\right)+m\left(0,0,2A.z+B.z\right)\right)
\operatorname{triangle}\left(A+\left(B.x+C.x,0,0\right)+v-m\left(C.x+B.x,0,0\right)+m\left(0,0,A.z+B.z\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)+v-m\left(C.x,0,0\right)+m\left(0,0,A.z+B.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)+v-m\left(C.x+B.x,0,0\right)+m\left(0,0,A.z\right)+mB+m\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(A+B+\left(C.x,0,0\right)+v-m\left(C.x,0,0\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)+v-m\left(C.x+B.x,B.y,0\right)+m\left(0,0,2A.z+B.z\right),A+\left(B.x+C.x,0,0\right)+v-m\left(C.x+B.x,0,0\right)\right)
v=-mA+mC
Folder: 1.2 STATIC
\operatorname{triangle}\left(A+B+\left(C.x,0,0\right),A+\left(B.x+C.x,0,0\right),A+B\right)
\operatorname{triangle}\left(A,A+\left(B.x+C.x,0,0\right),A+B\right)
\operatorname{triangle}\left(\left(A.x,\ A.y,0\right),A+B,\ A\right)
\operatorname{triangle}\left(\left(A.x,\ A.y,0\right),A+B,\ \left(\left(A+B\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(A+B+\left(C.x,0,0\right),A+B,\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(\left(\left(A+B\right).x,\left(A+B\right).y,0\right),A+B,\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right),A,A+\left(B.x+C.x,0,0\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right),A+\left(B.x+C.x,0,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right),\left(\left(A+B\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right),\left(\left(A+B\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(A+\left(B.x+C.x,0,0\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(A+\left(B.x+C.x,0,0\right),\left(\left(A+B+C\right).x,\left(A+B\right).y,0\right),A+B+\left(C.x,0,0\right)\right)
Folder: 2.1
\operatorname{triangle}\left(O,A,\left(A.x,\ A.y,0\right)\right)
\operatorname{triangle}\left(O,A,\left(A.x,0,A.z\right)\right)
\operatorname{triangle}\left(O,\left(A.x,0,A.z\right),\left(A.x,\ 0,0\right)\right)
\operatorname{triangle}\left(O,\left(A.x,\ A.y,0\right),\left(A.x,\ 0,0\right)\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right),\left(A.x,\ A.y,0\right),A\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right),\left(A.x,0,A.z\right),A\right)
Folder: 2.2
\operatorname{triangle}\left(A+B+C,B+C,B+C+\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(A+B+C,B+C,B+C+\left(0,A.y,0\right)\right)
\operatorname{triangle}\left(A+B+C,B+C+\left(0,0,A.z\right),B+C+\left(0,A.y,A.z\right)\right)
\operatorname{triangle}\left(A+B+C,B+C+\left(0,A.y,0\right),B+C+\left(0,A.y,A.z\right)\right)
\operatorname{triangle}\left(B+C+\left(0,A.y,A.z\right),B+C,B+C+\left(0,A.y,0\right)\right)
\operatorname{triangle}\left(B+C+\left(0,A.y,A.z\right),B+C,B+C+\left(0,0,A.z\right)\right)
Folder: 3.1
\operatorname{triangle}\left(\left(A.x,\ 0,0\right),\left(A.x,0,A.z\right),A\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right),\left(A.x,\ A.y,0\right),A\right)
\operatorname{triangle}\left(A,A+\left(B.x+C.x,0,0\right),\left(A.x,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right),A+\left(B.x+C.x,0,0\right),\left(A.x+B.x+C.x,0,A.z\right)\right)
\operatorname{triangle}\left(A-\left(0,0,A.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right),\left(A.x,0,A.z\right)-\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right)-\left(0,0,A.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right),\left(A.x+B.x+C.x,0,A.z\right)-\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right)+\left(B.x+C.x,0,0\right),\left(A.x,\ A.y,0\right)+\left(B.x+C.x,0,0\right),A+\left(B.x+C.x,0,0\right)\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right)+\left(B.x+C.x,0,0\right),\left(A.x,0,A.z\right)+\left(B.x+C.x,0,0\right),A+\left(B.x+C.x,0,0\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)-\left(0,A.y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,A.y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)-\left(0,A.y,0\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)-\left(0,A.y,0\right),A-\left(0,A.y,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,A.y,0\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right),A+\left(B.x+C.x,0,0\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right),A,A+\left(B.x+C.x,0,0\right)\right)
Folder: 3.2
\operatorname{triangle}\left(\left(A.x,\ 0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x,\ A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(A-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x+B.x+C.x,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(A-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x,0,A.z\right)-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right)-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x+B.x+C.x,0,A.z\right)-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right)+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x,\ A.y,0\right)+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,\ 0,0\right)+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),\left(A.x,0,A.z\right)+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)-\left(0,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)-\left(0,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)-\left(0,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A-\left(0,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-\left(0,0,A.z\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,A.y,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right),A+\left(B.x+C.x,0,0\right)-A+\left(0,A.y+B.y+C.y,A.z+B.z+C.z\right)\right)
Folder: 4.1
\operatorname{triangle}\left(O,C,\left(C.x,0,C.z\right)\right)
\operatorname{triangle}\left(O,C,\left(0,C.y,C.z\right)\right)
\operatorname{triangle}\left(O,\left(0,C.y,C.z\right),\left(0,0,C.z\right)\right)
\operatorname{triangle}\left(O,\left(0,0,C.z\right),\left(C.x,0,C.z\right)\right)
\operatorname{triangle}\left(\left(0,0,C.z\right),\left(C.x,0,C.z\right),C\right)
\operatorname{triangle}\left(\left(0,0,C.z\right),C,\left(0,C.y,C.z\right)\right)
Folder: 4.2
\operatorname{triangle}\left(A+B+C-mA,A+B-mA,A+B+\left(C.x,0,0\right)-mA\right)
\operatorname{triangle}\left(A+B-mB,A+B+\left(0,C.y,0\right)-mB,A+B+C-mB\right)
\operatorname{triangle}\left(A+B+C,A+B+\left(0,C.y,0\right),A+B+\left(C.x,C.y,0\right)\right)
\operatorname{triangle}\left(A+B+C,A+B+\left(C.x,0,0\right),A+B+\left(C.x,C.y,0\right)\right)
\operatorname{triangle}\left(A+B,A+B+\left(C.x,C.y,0\right),A+B+\left(C.x,0,0\right)\right)
\operatorname{triangle}\left(A+B,A+B+\left(C.x,C.y,0\right),A+B+\left(0,C.y,0\right)\right)
Folder: 5.1
\operatorname{triangle}\left(A+C,A+C+\left(0,0,B.z\right),A+B+C\right)
\operatorname{triangle}\left(A+C,A+C+\left(B.x,0,0\right),A+B+C\right)
\operatorname{triangle}\left(A+C+\left(0,0,B.z\right),A+B+C,A+C+\left(B.x,0,B.z\right)\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right),A+C+\left(B.x,\ 0,B.z\right),\ A+B+C\right)
\operatorname{triangle}\left(A+C,A+C+\left(0,0,B.z\right),\ A+C+\left(B.x,0,B.z\right)\right)
\operatorname{triangle}\left(A+C,A+C+\left(B.x,0,B.z\right),A+C+\left(B.x,0,0\right)\right)
Folder: 5.2
\operatorname{triangle}\left(O,B,\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(O,B,\left(B.x,B.y,0\right)\right)
\operatorname{triangle}\left(O,\left(B.x,B.y,0\right),\left(0,B.y,0\right)\right)
\operatorname{triangle}\left(O,\left(0,B.y,0\right),\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(\left(0,B.y,0\right),\left(0,B.y,B.z\right),B\right)
\operatorname{triangle}\left(\left(0,B.y,0\right),\left(B.x,B.y,0\right),B\right)
Folder: 6.1
\operatorname{triangle}\left(C,\left(C.x,0,C.z\right),\ \left(0,C.y,C.z\right)\right)
\operatorname{triangle}\left(\left(0,0,C.z\right),\left(C.x,0,C.z\right),\ \left(0,C.y,C.z\right)\right)
\operatorname{triangle}\left(C+\left(0,0,A.z+B.z\right),\left(C.x,0,C.z\right)+\left(0,0,A.z+B.z\right),\ \left(0,C.y,C.z\right)+\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right),\left(C.x,0,C.z\right)+\left(0,0,A.z+B.z\right),\ \left(0,C.y,C.z\right)+\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(0,0,C.z\right),\left(0,C.y,C.z\right),\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+B.z+A.z\right),\left(0,C.y,C.z\right),\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+\left(C.x,0,0\right),\left(0,C.y,C.z\right)+\left(C.x,0,0\right),\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+\left(C.x,0,0\right)\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+B.z+A.z\right)+\left(C.x,0,0\right),\left(0,C.y,C.z\right)+\left(C.x,0,0\right),\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+\left(C.x,0,0\right)\right)
\operatorname{triangle}\left(C,C+\left(0,0,A.z+B.z\right),\left(0,C.y,C.z+A.z+B.z\right)\right)
\operatorname{triangle}\left(C,\left(0,C.y,C.z\right),\left(0,C.y,C.z+A.z+B.z\right)\right)
\operatorname{triangle}\left(C-\left(0,C.y,0\right),C+\left(0,0,A.z+B.z\right)-\left(0,C.y,0\right),\left(0,C.y,C.z+A.z+B.z\right)-\left(0,C.y,0\right)\right)
\operatorname{triangle}\left(C-\left(0,C.y,0\right),\left(0,C.y,C.z\right)-\left(0,C.y,0\right),\left(0,C.y,C.z+A.z+B.z\right)-\left(0,C.y,0\right)\right)
Folder: 6.2
v_{1}=-C+\left(A.x+B.x+C.x,A.y+B.y+C.y,0\right)
\operatorname{triangle}\left(C+v_{1},\left(C.x,0,C.z\right)+v_{1},\ \left(0,C.y,C.z\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+v_{1},\left(C.x,0,C.z\right)+v_{1},\ \left(0,C.y,C.z\right)+v_{1}\right)
\operatorname{triangle}\left(C+\left(0,0,A.z+B.z\right)+v_{1},\left(C.x,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1},\ \left(0,C.y,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1},\left(C.x,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1},\ \left(0,C.y,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+v_{1},\left(0,C.y,C.z\right)+v_{1},\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+B.z+A.z\right)+v_{1},\left(0,C.y,C.z\right)+v_{1},\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+\left(C.x,0,0\right)+v_{1},\left(0,C.y,C.z\right)+\left(C.x,0,0\right)+v_{1},\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+\left(C.x,0,0\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+B.z+A.z\right)+\left(C.x,0,0\right)+v_{1},\left(0,C.y,C.z\right)+\left(C.x,0,0\right)+v_{1},\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+\left(C.x,0,0\right)+v_{1}\right)
\operatorname{triangle}\left(C+v_{1},C+\left(0,0,A.z+B.z\right)+v_{1},\left(0,C.y,C.z+A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(C+v_{1},\left(0,C.y,C.z\right)+v_{1},\left(0,C.y,C.z+A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(C-\left(0,C.y,0\right)+v_{1},C+\left(0,0,A.z+B.z\right)-\left(0,C.y,0\right)+v_{1},\left(0,C.y,C.z+A.z+B.z\right)-\left(0,C.y,0\right)+v_{1}\right)
\operatorname{triangle}\left(C-\left(0,C.y,0\right)+v_{1},\left(0,C.y,C.z\right)-\left(0,C.y,0\right)+v_{1},\left(0,C.y,C.z+A.z+B.z\right)-\left(0,C.y,0\right)+v_{1}\right)
Folder: 7.1
\operatorname{triangle}\left(A+C,A+C+\left(0,0,B.z\right),A+C+\left(B.x,0,B.z\right)\right)
\operatorname{triangle}\left(A+C,A+C+\left(B.x,0,B.z\right),A+C+\left(B.x,0,0\right)\right)
\operatorname{triangle}\left(A+C,\left(A.x+C.x,0,A.z+C.z\right),\left(A.x+C.x,0,A.z+C.z+B.z\right)\right)
\operatorname{triangle}\left(A+C,A+C+\left(0,0,B.z\right),\left(A.x+C.x,0,A.z+C.z+B.z\right)\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right),\left(A.x+C.x,0,A.z+C.z\right)+\left(B.x,0,0\right),\left(A.x+C.x,0,A.z+C.z+B.z\right)+\left(B.x,0,0\right)\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right),A+C+\left(0,0,B.z\right)+\left(B.x,0,0\right),\left(A.x+C.x,0,A.z+C.z+B.z\right)+\left(B.x,0,0\right)\right)
\operatorname{triangle}\left(A+C-\left(0,A.y+C.y,0\right),A+C+\left(0,0,B.z\right)-\left(0,A.y+C.y,0\right),A+C+\left(B.x,0,B.z\right)-\left(0,A.y+C.y,0\right)\right)
\operatorname{triangle}\left(A+C-\left(0,A.y+C.y,0\right),A+C+\left(B.x,0,B.z\right)-\left(0,A.y+C.y,0\right),A+C+\left(B.x,0,0\right)-\left(0,A.y+C.y,0\right)\right)
\operatorname{triangle}\left(A+C,A+C+\left(B.x,0,0\right),A+C+\left(B.x,-C.y-A.y,0\right)\right)
\operatorname{triangle}\left(A+C,\left(A.x+C.x,0,C.z+A.z\right),A+C+\left(B.x,-C.y-A.y,0\right)\right)
\operatorname{triangle}\left(A+C+\left(0,0,B.z\right),A+C+\left(B.x,0,0\right)+\left(0,0,B.z\right),A+C+\left(B.x,-C.y-A.y,0\right)+\left(0,0,B.z\right)\right)
\operatorname{triangle}\left(A+C+\left(0,0,B.z\right),\left(A.x+C.x,0,C.z+A.z\right)+\left(0,0,B.z\right),A+C+\left(B.x,-C.y-A.y,0\right)+\left(0,0,B.z\right)\right)
Folder: 7.2
v_{2}=-\left(A+C\right)+\left(0,B.y+C.y+A.y,0\right)
\operatorname{triangle}\left(A+C+v_{2},A+C+\left(0,0,B.z\right)+v_{2},A+C+\left(B.x,0,B.z\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+v_{2},A+C+\left(B.x,0,B.z\right)+v_{2},A+C+\left(B.x,0,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+v_{2},\left(A.x+C.x,0,A.z+C.z\right)+v_{2},\left(A.x+C.x,0,A.z+C.z+B.z\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+v_{2},A+C+\left(0,0,B.z\right)+v_{2},\left(A.x+C.x,0,A.z+C.z+B.z\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right)+v_{2},\left(A.x+C.x,0,A.z+C.z\right)+\left(B.x,0,0\right)+v_{2},\left(A.x+C.x,0,A.z+C.z+B.z\right)+\left(B.x,0,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right)+v_{2},A+C+\left(0,0,B.z\right)+\left(B.x,0,0\right)+v_{2},\left(A.x+C.x,0,A.z+C.z+B.z\right)+\left(B.x,0,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C-\left(0,A.y+C.y,0\right)+v_{2},A+C+\left(0,0,B.z\right)-\left(0,A.y+C.y,0\right)+v_{2},A+C+\left(B.x,0,B.z\right)-\left(0,A.y+C.y,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C-\left(0,A.y+C.y,0\right)+v_{2},A+C+\left(B.x,0,B.z\right)-\left(0,A.y+C.y,0\right)+v_{2},A+C+\left(B.x,0,0\right)-\left(0,A.y+C.y,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+v_{2},A+C+\left(B.x,0,0\right)+v_{2},A+C+\left(B.x,-C.y-A.y,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+v_{2},\left(A.x+C.x,0,C.z+A.z\right)+v_{2},A+C+\left(B.x,-C.y-A.y,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+\left(0,0,B.z\right)+v_{2},A+C+\left(B.x,0,0\right)+\left(0,0,B.z\right)+v_{2},A+C+\left(B.x,-C.y-A.y,0\right)+\left(0,0,B.z\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+\left(0,0,B.z\right)+v_{2},\left(A.x+C.x,0,C.z+A.z\right)+\left(0,0,B.z\right)+v_{2},A+C+\left(B.x,-C.y-A.y,0\right)+\left(0,0,B.z\right)+v_{2}\right)
Folder: 8.1
\operatorname{triangle}\left(C,C+\left(0,0,A.z+B.z\right),A+C\right)
\operatorname{triangle}\left(A+C,A+C+\left(0,0,B.z\right),C+\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(\left(C.x,0,C.z\right),\left(C.x,0,C.z+B.z+A.z\right),C+\left(0,0,A.z+B.z\right)\right)
\operatorname{triangle}\left(C,C+\left(0,0,A.z+B.z\right),\left(C.x,0,C.z\right)\right)
\operatorname{triangle}\left(A+C,A+C+\left(0,0,B.z\right),\left(\left(A+C\right).x,0,\left(A+C\right).z\right)\right)
\operatorname{triangle}\left(\left(\left(A+C\right).x,0,\left(A+C\right).z+B.z\right),A+C+\left(0,0,B.z\right),\left(\left(A+C\right).x,0,\left(A+C\right).z\right)\right)
\operatorname{triangle}\left(A+C-\left(0,A.y+C.y,0\right),\left(C.x,0,C.z\right),\left(C.x,0,C.z+A.z+B.z\right)\right)
\operatorname{triangle}\left(A+C-\left(0,A.y+C.y,0\right),\left(A.x+C.x,0,A.z+C.z+B.z\right),\left(C.x,0,C.z+A.z+B.z\right)\right)
\operatorname{triangle}\left(C,A+C,\left(A.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(C,\left(C.x,0,C.z\right),\left(A.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(C+\left(0,0,A.z+B.z\right),A+C+\left(0,0,B.z\right),\left(A.x+C.x,0,A.z+C.z+B.z\right)\right)
\operatorname{triangle}\left(C+\left(0,-C.y,A.z+B.z\right),C+\left(0,0,A.z+B.z\right),\left(A.x+C.x,0,A.z+C.z+B.z\right)\right)
Folder: 8.2
\operatorname{triangle}\left(A+B+\left(0,C.y,0\right),B,\left(B.x,\ B.y+A.y+C.y,B.z\right)\right)
\operatorname{triangle}\left(A+B,A+B+\left(0,C.y,0\right),B\right)
\operatorname{triangle}\left(\left(0,C.y,C.z+B.z+A.z\right)+v_{1},\left(0,C.y,C.z\right)+v_{1},\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(\left(0,0,C.z\right)+v_{1},\left(0,C.y,C.z\right)+v_{1},\left(0,0,C.z\right)+\left(0,0,A.z+B.z\right)+v_{1}\right)
\operatorname{triangle}\left(A+B,\left(B.x,\ B.y,0\right),B\right)
\operatorname{triangle}\left(A+B,\left(B.x,\ B.y,0\right),\left(\left(A+B\right).x,\left(A+B\right).y,0\right)\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right)+v_{2},A+C+\left(0,0,B.z\right)+\left(B.x,0,0\right)+v_{2},\left(A.x+C.x,0,A.z+C.z+B.z\right)+\left(B.x,0,0\right)+v_{2}\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right)+v_{2},\left(A.x+C.x,0,A.z+C.z\right)+\left(B.x,0,0\right)+v_{2},\left(A.x+C.x,0,A.z+C.z+B.z\right)+\left(B.x,0,0\right)+v_{2}\right)
\operatorname{triangle}\left(\left(B.x,B.y,0\right),\left(B.x,B.y+A.y+C.y,0\right),\left(A.x+B.x,A.y+B.y,0\right)\right)
\operatorname{triangle}\left(\left(B.x+A.x,B.y+A.y+C.y,0\right),\left(B.x,B.y+A.y+C.y,0\right),\left(A.x+B.x,A.y+B.y,0\right)\right)
\operatorname{triangle}\left(\left(B.x+A.x,B.y+A.y+C.y,0\right),A+B+\left(0,C.y,0\right),B+\left(0,C.y+A.y,0\right)\right)
\operatorname{triangle}\left(\left(B.x,B.y+A.y+C.y,0\right),A+B+\left(0,C.y,0\right)-\left(0,0,A.z+B.z\right),B+\left(0,C.y+A.y,0\right)\right)
Folder: 9.1
\operatorname{triangle}\left(A,\left(A.x,0,A.z\right),\left(A.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x+C.x,0,A.z+C.z\right),A+C,A\right)
\operatorname{triangle}\left(A+C,\left(A.x+C.x,0,C.z+A.z\right),A+C+\left(B.x,-C.y-A.y,0\right)\right)
\operatorname{triangle}\left(A+C,A+C+\left(B.x,0,0\right),A+C+\left(B.x,-C.y-A.y,0\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right),A+\left(B.x+C.x,0,0\right),\left(A.x+B.x+C.x,0,A.z\right)\right)
\operatorname{triangle}\left(A,A+\left(B.x+C.x,0,0\right),\left(A.x,0,A.z\right)\right)
\operatorname{triangle}\left(A+C,A,A+C+\left(B.x,0,0\right)\right)
\operatorname{triangle}\left(A,A+\left(B.x+C.x,0,0\right),A+C+\left(B.x,0,0\right)\right)
\operatorname{triangle}\left(A+C+\left(B.x,0,0\right),A+\left(B.x+C.x,0,0\right),\left(C.x+B.x+A.x,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right),\left(A.x+C.x,0,A.z+C.z\right),\left(A.x+B.x+C.x,0,A.z+C.z\right)\right)
\operatorname{triangle}\left(\left(A.x,0,A.z\right),\left(A.x+C.x+B.x,0,A.z+C.z\right),\left(A.x+B.x+C.x,0,A.z\right)\right)
\operatorname{triangle}\left(\left(A.x+C.x+B.x,0,A.z+C.z\right),A+C+\left(B.x,0,0\right),\left(C.x+B.x+A.x,0,A.z\right)\right)
Folder: 9.2
\operatorname{triangle}\left(B+C,\ \left(0,B.y+C.y,B.z+C.z\right),\left(0,B.y+C.y+A.y,B.z+C.z\right)\right)
\operatorname{triangle}\left(B+C,\left(0,B.y+C.y+A.y,B.z+C.z\right),B+C+\left(0,A.y,0\right)\right)
\operatorname{triangle}\left(B+C+\left(0,A.y,0\right),\left(0,B.y+C.y+A.y,B.z+C.z\right),B+\left(0,C.y+A.y,0\right)\right)
\operatorname{triangle}\left(B+C+\left(0,A.y,0\right),B+C,B+\left(0,A.y+C.y,0\right)\right)
\operatorname{triangle}\left(B+C,B,B+\left(0,A.y+C.y,0\right)\right)
\operatorname{triangle}\left(\left(0,\left(B+C\right).y,\left(B+C\right).z\right),B+C,\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(B+C,\left(0,B.y,B.z\right),B\right)
\operatorname{triangle}\left(\left(0,B.y+C.y,B.z+C.z\right),\left(0,B.y+C.y+A.y,B.z+C.z\right),\left(0,B.y,B.z\right)\right)
\operatorname{triangle}\left(\left(0,B.y,B.z\right),\left(0,B.y+A.y+C.y,B.z\right),\left(0,B.y+A.y+C.y,B.z+C.z\right)\right)
\operatorname{triangle}\left(\left(0,B.y+A.y+C.y,B.z+C.z\right),B+\left(0,C.y+A.y,0\right),\left(0,B.y+C.y+A.y,B.z\right)\right)
\operatorname{triangle}\left(B,B+\left(0,C.y+A.y,0\right),\left(0,B.y+C.y+A.y,B.z\right)\right)
\operatorname{triangle}\left(B,\left(0,B.y+C.y+A.y,B.z\right),\left(0,B.y,B.z\right)\right)

Теги:

Хабы:

Визуализация чисто геометрического вывода формулы определителя третьего порядка

Геометрический вывод в 2D — вычисление (ориентированной) площади серого параллелограмма на картинке:

Переход в 3D

Ссылки на Desmos и разметка:

Публикации

Истории

Ближайшие события