The Pythagorean Theorem: The Great Deception of the School Curriculum. How Abstraction Killed Meaning / Habr

We all know this formula .

This is perhaps the only piece of knowledge from school geometry that stays with a person for their entire life, even if they work as a barista or a courier.

But have you ever asked yourself the question: why squares specifically?
Why not cubes? Why not just the sum of the absolute values ?

If you ask a teacher, they will draw little squares on the sides of the triangle. If you ask a university professor, they will write down the definition of a scalar product.
And both of them, in essence, will deceive you. Or, to put it more mildly, they won't tell you the whole truth.

Today, we will unpack this 'black box' and see that the Pythagorean theorem is not about triangles at all. And it should be proven in a completely different way than we were taught.

The school curriculum doesn't provide an answer. Moreover, the history of teaching the Pythagorean theorem is a story of how living, visual geometry was turned into dry, dead algebra. We were led further and further away from understanding the essence and towards abstraction.

Today, we will analyze this path of degradation and show a proof that will bring you back to reality. Spoiler: the Pythagorean theorem is not about triangles. It's about mirrors.

Get ready to have all your preconceptions shattered!

A Chronicle of Lost Meaning: From Mechanics to Abstraction

I myself graduated from a regular high school in 2005. I remember that my classmates had absolutely no understanding of the proof of the Pythagorean theorem and couldn't reproduce it because it required an additional construction (drawing an altitude), comparing pairs of similar triangles, and performing calculations. It's a dry, abstract proof with no clear meaning behind it. But why is this the one they teach?

Let's look at how the proofs in textbooks have changed. Instead of making understanding simpler, education has made it increasingly abstract and detached from the physical world, and less and less comprehensible to schoolchildren and students.

1. The Golden Age: 'Pythagorean Trousers' and Liquid Geometry

In classical geometry (for example, in Euclid's 'Elements'), the proof was a masterpiece of engineering thought. It was based on shearing.

How it works.

Imagine a square built on a leg of the triangle. Mentally 'push' its top side parallel to the base. The square will turn into a parallelogram.

Пифагоровы штаны во все стороны равны — Pythagorean trousers are equal on all sides

Physics: imagine a deck of cards. If you slide the deck sideways, its shape will change, but the area of the side face will remain the same. The height is the same, the base is the same.
Next, this parallelogram is 'shifted' and rotated downwards, turning into a rectangle that perfectly fills part of the square on the hypotenuse.
The drawing itself, with three squares extending in different directions, reminded schoolchildren of the cut of men's trousers. This is where the famous saying 'Pythagorean trousers are equal on all sides' came from, which was memorized by generations, often without even understanding the essence of the proof. And nowadays, very few people know its meaning.

I'll add another animation variant that shows the descent stage a bit more clearly (but one could also refer to the formula for the area of a triangle.

Второй вариант анимации — Second animation variant

It was a proof-cartoon.

You could see how the area of one square flowed, like a liquid, into another, changing its shape but preserving its volume. It was absolutely rigorous and yet intuitively understandable to anyone who had ever seen a crooked fence.

Why it was removed:

because a trend emerged towards the arithmetization of analysis and the exclusion of geometric considerations from academic courses. Living geometry was replaced by dead letters.

Why does it work? (The magic of shearing)

In the illustration, we see an intermediate stage: the squares have turned into two parallelograms. Two questions arise for the reader (and for the attentive critic):

Why are we sure the area hasn't changed?
Why do these parallelograms fit together so perfectly along their long sides, forming a single figure?

Let's prove this visually and rigorously.

1. Area Invariance

Take a look at the purple parallelogram.

Its base is the leg of the original triangle (length ). It is stationary.
Its height is the distance between the parallel lines (between the leg and the top side of the original square). This distance is equal to the side of the square ().

Площадь при перекашивании не меняется. — The area does not change during shearing.

During the shear, the top edge slides along a parallel line.

$S = \text{Основание} \times \text{Высота}$

Since neither the base (), nor the height () have changed, the area of the parallelogram is identically equal to the area of the original square ().

The same is true for the green one ().

2. Why do they meet at one point? (the hidden rotation)

This is the most beautiful part. Look at the long, slanted sides of the blue and green parallelograms. What is their length?

Imagine the triangle that is 'cut off' by the shear from the bottom and 'glued' on at the top.
This small triangle is congruent to our original right-angled triangle!

During the shear, we deform the square until the side face becomes vertical (perpendicular to the hypotenuse, if you flip the picture).
Geometrically, this means that the long slanted side of the parallelogram is equal to the hypotenuse () of the original triangle.

Proof:
If we rotate the original triangle by 90 degrees, its legs will become the sides of our squares, and the hypotenuse will become precisely that long side of the parallelogram.

Тайна скрытого поворота — The secret of the hidden rotation

Общая схема на одном рисунке — Overall diagram in one drawing

Можно сделать из бумаги. — Can be made from paper.

Conclusion:

The long side of the blue parallelogram is equal to .
The long side of the green parallelogram is equal to .
Therefore, when they 'flow' down (in the second phase of the animation), they form a square with side .

And the vertical line where they touch in the drawing is precisely the altitude dropped from the right angle to the hypotenuse. It divides the square on the hypotenuse into two rectangles with areas equal to and respectively.

2. A Step Towards Degradation: The Puzzle Method (Areas)

The first stage of degradation is the dissection method. The one where a square is cut into pieces (usually 4 triangles and a small square) and they are rearranged.

Метод перекладывания — The rearrangement method

'Behold!': The Boldest Gesture in the History of Mathematics

This picture has a surprising history.

In the 12th century, the great Indian mathematician Bhaskara II (also known as Bhaskaracharya) included this diagram in his monumental work 'Crown of Learning' (Lilavati). Imagine: you are reading a serious scientific treatise. There are complex calculations, astronomical tables... And suddenly—a page with no axioms, no theorems, no long equations.

Bhaskara drew a square with four triangles inside it, and simply wrote one word in Sanskrit underneath:

'Behold!'

Bhaskara understood a fundamental thing that we have forgotten: geometry is the art of seeing, not the art of juggling symbols. If you can see how four triangles regroup, leaving different 'holes', any words become informational noise. The proof was completed in your brain the moment you simply looked at the drawing.

The Chinese Connection: Gou and Gu

But even Bhaskara was not the pioneer.

A thousand years before him, when Pythagoras may not have even been born, Chinese astronomers were already using this diagram as a self-evident fact.

In the ancient treatise 'Zhoubi Suanjing' (a mathematical treatise on the gnomon of the Zhou, c. 1st century BC), this theorem was called the Gougu theorem. The Chinese thought in concrete terms:

Gou (hook) is the shorter leg.
Gu (thigh) is the longer leg.
Xian (chord) is the hypotenuse.

For ancient engineers and astronomers, the truth of the formula was as obvious as the fact that water is wet. They didn't need proofs using abstract triangles because they used this principle every day to measure fields and observe the stars.

What's the catch: yes, it's visual, like a child's puzzle. But it kills the understanding of continuity of space. This is discrete geometry. We see that it 'fits', but we don't feel the law by which it fits. It's a magic trick, not science.

But even this proof began to disappear from school textbooks.

The tragedy is this: we replaced the crystal-clear, intuitive 'Behold!' with pages of dry calculations. We stopped trusting our eyes and started believing only in formulas. And it was at this moment that the Pythagorean theorem turned from living knowledge into a boring 'chore' for passing exams.

3. A Complete Break with Reality: Similar Triangles

And finally, the modern rock bottom of the school curriculum—the proof using similar triangles. We draw an altitude, write out proportions like , cross-multiply... and get the formula.

Abstraction takes over

Look at how it's taught now. The areas have disappeared, the 'liquid geometry' has disappeared. Only dry lines and proportions remain.

Рисунок к теореме Пифагора — Diagram for the Pythagorean theorem

The proof looks impeccably rigorous, but it completely hides the essence of what's happening behind a thicket of formulas and is not actually rigorous.

Let $\angle B = 90^\circ$ . Let's draw the altitude (the orange line).
Let's consider the angles. In a right-angled triangle, the sum of the acute angles is $90^\circ$ .
- $\angle A + \angle C = 90^\circ$
- $\angle CBH + \angle C = 90^\circ$ (in triangle )
- Therefore, $\angle A = \angle CBH$ (marked with orange arcs).
The magic of similarity:
From the equality of angles, it follows that the small triangle is similar to the large one .
Let's set up a proportion (the ratio of a leg to the hypotenuse):
$\frac{HC}{BC} = \frac{BC}{AC} \quad \Rightarrow \quad BC^2 = AC \cdot HC \quad (1)$
Similarly, triangle is similar to the large one .
$\frac{AH}{AB} = \frac{AB}{AC} \quad \Rightarrow \quad AB^2 = AC \cdot AH \quad (2)$
The algebraic finale:
Simply add equations (1) and (2):
Factor out the common term :
$AB^2 + BC^2 = AC \cdot (AH + HC)$
But is the entire hypotenuse .
$AB^2 + BC^2 = AC \cdot AC = AC^2$

What's the catch?

There are two problems here: one is obvious, and the second is fundamental, one that even teachers keep silent about.

1. The Problem of Visualization: The 'Black Box'
This proof works like a meat grinder: you load triangles into the input, turn the crank of algebra (proportions), and out comes the minced meat of the formula .
But you don't see why this happened. Where are the squares here? How do they physically assemble? You only see the arithmetic of letters. The geometric meaning is completely lost.

2. The Problem of Rigor: The Great School Deception
We are told that the proof through similarity is the most rigorous and scientific one. This is a lie.

Intuitively, similarity is understandable—it's just 'zooming' a figure. But strictly mathematically, similarity relies on Thales's theorem about proportional segments. And that's where the catch is.

In textbooks, Thales's theorem is only honestly proven for commensurable segments (whose lengths have a rational ratio, i.e., a fraction).
But in a right-angled triangle, the sides are often incommensurable (if the legs are 1 and 1, then the hypotenuse is $\sqrt{2}$ — an irrational number).

To honestly prove similarity for irrational numbers, you need the theory of limits or the theory of real numbers (Dedekind cuts). This is not in the 8th-grade curriculum.

What do textbook authors do?

They commit a logical crime: they prove the theorem for the simple case (fractions), and then silently, 'by contraband', apply it to all cases.

The result.

The school proof using similarity is a colossus with feet of clay. It not only kills geometric intuition but is also logically 'full of holes' itself, creating in children an illusion of rigor where there is none. We have been taught to blindly believe in formulas, even if the foundation beneath them is built on empty words.

3. The University 'Cheat Code': Vector Algebra

Entering a technical university, a student thinks, 'Well, now they'll explain everything to me like an adult.'
In a linear algebra lecture, a professor comes out and writes:

> Definition: The scalar product of vectors $\vec{a}(x_1, y_1)$ and $\vec{b}(x_2, y_2)$ is defined as the number .
> Definition: The length of a vector is defined as $|\vec{a}| = \sqrt{(\vec{a}, \vec{a})} = \sqrt{x^2 + y^2}$ .

Stop. Wait a minute.

You just postulated the Pythagorean theorem inside a definition!

The main problem: we were given a formula but not its physical meaning. Why is length the square root of the sum of squares? Why is the metric of our world precisely this way? The vector definition is a convenient tool for an engineer, but it's a surrender for a mathematician who has stopped asking 'why'.

The natural approach should be different—to derive the Pythagorean theorem from the familiar properties of space and our understanding of length.

The most unexpected thing is that this approach is not only very visual but is also even more natural, understandable, and profound than Euclid's approach!

4. The Proof of the Pythagorean Theorem from The Book

The famous mathematician Paul Erdős often referred to 'The Book' as the place where God keeps the best proofs of mathematical theorems. In a 1985 lecture, Erdős said, 'You don't have to believe in God, but you should believe in The Book.'

The best means simple, beautiful, conceptual proofs that provide an understanding of the true reasons for the validity of a mathematical fact, without resorting to excesses.

But which proof of the Pythagorean theorem is written in 'The Book'?

To find it, let's forget everything we were taught. Forget about areas, triangles, and scalar products. Let's leave only bare logic and the properties of axial symmetry.

All we need to know about our space is that it is symmetric.

If you look in a mirror, your reflection will have the same dimensions as you. Length does not change upon reflection. This is enough to derive the Pythagorean theorem.

Step 1. The Mirror Experiment

Let's take a vector lying on the X-axis. Let its coordinates be $\vec{A} = (L; 0)$ . Its length is .
Now let's reflect this vector across some inclined axis (a mirror) passing through the origin. We will get a new vector $\vec{B} = (x; y)$ .

Зеркальное отражение вектора — Mirror reflection of a vector

Key fact: since this is a reflection, the length of the vector has not changed.

Our task is to find the relationship between and .

Step 2. Constructing a Rhombus

Let's look at our two vectors: the original $\vec{A}$ and the reflected $\vec{B}$ . Let's construct a rhombus on them. Any rhombus has two diagonals, and they have a remarkable property:

The sum vector ( $\vec{S} = \vec{B} + \vec{A}$ ) is the major diagonal. It coincides with the axis of symmetry (our mirror).
The difference vector ( $\vec{D} = \vec{B} - \vec{A}$ ) is the minor diagonal. It connects the ends of the vectors.

Look at the picture (in your mind or on the screen): in a rhombus, the diagonals are always perpendicular. The mirror ( $\vec{S}$ ) is perpendicular to the plane of reflection ( $\vec{D}$ ).

Схема отражения, диагонали перпендикулярны — Reflection diagram, diagonals are perpendicular

Step 3. The Algebra of Slopes (without the scalar product!)

We know the coordinates of the vectors:

$\vec{A} = (L; 0)$
$\vec{B} = (x; y)$

Let's find the coordinates of the diagonals:

Sum $\vec{S} = (x + L;\; y)$
Difference $\vec{D} = (x - L;\; y)$

And now let's use the property of perpendicularity, which is known even to schoolchildren who haven't heard of vectors but have solved problems on 'graph paper'.

If two lines are perpendicular, then the product of their slopes (gradient coefficients $k = \Delta y / \Delta x$ ) is equal to -1.

(This is easy to understand: a 90-degree rotation swaps and and changes the sign of one of the coordinates. The slope was , it became . Their product gives ).

Let's write this down for our diagonals:

Slope of the sum: $k_1 = \frac{y}{x + L}$
Slope of the difference: $k_2 = \frac{y}{x - L}$

Multiplying them:

$\frac{y}{x + L} \cdot \frac{y}{x - L} = -1$

🤓 For pedants and mathematicians: isn't there a vicious circle here?

An attentive reader might exclaim:

'Wait! You are using the fact that the product of the slopes of perpendicular lines is . But isn't this property derived from the very same Pythagorean theorem or the scalar product? Aren't we just proving a tautology?'

Answer: no. There is no vicious circle here.

It all depends on what we choose as our foundation (axiomatics).
In vector algebra, one starts with lengths (metrics), and angles and rotations are derived from them.

We, however, are taking the opposite, more modern path (in the spirit of Felix Klein's 'Erlangen Program'):

We postulate that space is homogeneous and isotropic.
We define a rotation by $90^\circ$ as a linear transformation that maps the axis to the axis, and the axis to the axis. This is a purely algebraic fact: the vector is transformed into . You don't need to know the metric for this. Graph paper is sufficient. We define the coordinates of points using these same grid cells.
Slope of the original vector: .
Slope of the rotated vector: .
Their product is equal to minus one.

We did not use the Pythagorean theorem to get . We used the symmetry properties of a right-angle rotation.

And from this property, we derive that the only metric that is preserved under such rotations is the sum of squares ().

We are not forcing the answer. We are showing that the quadratic metric is an inevitable consequence of the symmetry of our space.

Step 4. The Emergence of Squares

Look what happens.

The squares appear not artificially, but naturally, from algebraic multiplication!

In the numerator: $y \cdot y = y^2$ .

In the denominator: the difference of squares formula .

We get the equation:

$\frac{y^2}{x^2 - L^2} = -1$

We get rid of the fraction (multiply by the denominator):

$y^2 = -1 \cdot (x^2 - L^2)$

We expand the parenthesis (the minus sign changes the signs):

We move to the left:

That's it.

5. What did we just do?

We proved the Pythagorean theorem without drawing a single right-angled triangle.
We didn't calculate the areas of squares.
We didn't use dubious similarity.
We didn't postulate the scalar product.

We used only one property of space: symmetry under reflection. And the squares in the formula are an inevitable consequence of how the 'forward' slope and the 'backward' slope interact under perpendicularity.

This method shows the true nature of things: is a law of conservation of information when rotating or reflecting the coordinate grid.

And this is exactly how this theorem should be understood in the 21st century.

🎁 Bonus Level: A Greeting from Thales

The attentive reader might have noticed something familiar in our recent calculations. Let's look at the slope equation again:

$\frac{y}{x+L} \cdot \frac{y}{x-L} = -1$

What is this from the perspective of classical geometry?
Look at the denominators: and . This is a hint at two points on the X-axis:

Point is the left end of a circle's diameter.
Point is the right end of the diameter.

Our fractions are not just abstract numbers.

$\frac{y}{x+L}$ is the slope of the line connecting our point with the left end of the diameter .
$\frac{y}{x-L}$ is the slope of the line connecting point with the right end of the diameter .

Their product is minus one. This means that these lines are perpendicular.

We have just, without realizing it, rediscovered Thales's theorem: 'An angle inscribed in a semicircle is a right angle.'

Our 'Mirror' (the sum of vectors) is parallel to one chord.
Our 'Difference Vector' is parallel to the second chord.

The circle is complete. The algebraic property of the perpendicularity of a rhombus's diagonals and the geometric property of points on a circle are one and the same. The formula is simply the analytical notation of the fact that we are on a circle.

Теорема Фалеса о вписанном угле — Thales's theorem on the inscribed angle

Global Conclusion: A Mirror is All You Need

What we did today with the Pythagorean theorem is just the tip of the iceberg.

In school, geometry is taught as a collection of disparate facts: here are the criteria for triangle congruence, here are the properties of a parallelogram, here is the tangent to a circle. It seems like a hundred different rules and proofs that need to be memorized.

But in reality, all of school plane geometry (and a large part of solid geometry) can be derived from a single principle—Axial Symmetry.

Isosceles triangle? That's symmetry.
Rhombus? Two symmetries.
Kite? One symmetry.
Circle? An infinite number of symmetries.
The properties of tangents, angle bisectors, perpendicular bisectors? All of these are just consequences of reflection in a mirror.

If a geometry course is built not on Euclid's axioms, but on transformations (motion and reflection), the subject becomes many times simpler, more visual, and, most importantly, more logical. You no longer need to cram proofs—you see them.

I will definitely write a series of articles on how to reassemble all of school geometry based on mirrors and rotations. But before that, something more large-scale awaits us.

First, we will deal with Calculus. I will show how to understand and rigorously derive all the theorems without the torment caused by standard textbooks.

And then, based on our new metric and understanding of curvature, we will take a swing at the holy of holies of physics—the General Theory of Relativity. And you will be surprised at how simple and understandable it will become if you approach it with the right geometric keys.

Subscribe, leave a like, so you don't miss out. It's just the beginning.

And now for the homework assignment for the bravest among you: try to apply the reflection method not to a right-angled triangle, but to an arbitrary one. You will see how the term $-2 a b \cos \alpha$ . Write your versions of a simple and elegant proof of the law of cosines in the comments.