Euclid’s theorems

The ancient Greek mathematician Euclid of Alexandria set out two theorems about triangles, specifically about the altitude and the shorter sides (the legs).

Euclid’s altitude theorem

This theorem works only in a right-angled triangle. Let’s look at the basic set-up:

Here we have triangle ABC, where side c is the hypotenuse and the segment h_c is the altitude (perpendicular height) to side c. The point P_c splits side c into two segments: we label the segment AP_c as c_b and the segment BP_c as c_a.

Euclid’s altitude theorem then says that

$$h_c^2=c_a\cdot c_b,$$

where h_c stands for the length of the segment h_c, and so on. In other words: if we built a square with side length h_c and a rectangle with side lengths c_a and c_b, these two shapes would have the same area. Have a look at the following picture, in which exactly such a square and rectangle are highlighted:

Euclid’s altitude theorem says that these shapes have the same area. The red square has side length h_c, the green rectangle has one side of length c_a and the other of length c_b.

You can read a proof of the altitude theorem on Wikipedia.

Euclid’s leg theorem

This second theorem of Euclid also holds in a right-angled triangle. We stay with the first picture of the triangle:

The second theorem states that

$$\begin{array}{rcl}{a}^2& =& c\cdot c_a\\ b^2& =& c\cdot c_b\end{array}$$

If we build a square with side length a, this square has the same area as a rectangle with side lengths c and c_a. Once again, you can look at the square and the rectangle:

You can find the proof on Wikipedia again.