Euclid's theorems

The ancient Alexandrian mathematician Euclid composed two theorems concerning triangles, specifically concerning height and the hypotenuse.

Euclid's theorem on height

This theorem only works on right triangles. Let's look at the basic assignment:

Here we have a triangle ABC, side c is the hypotenuse, and the line segment v_c is the height to side c. The point P_c divides the side c into two segments: we label the line segment AP_c c_b , and we label the line segment BP_c c_a .

Euclid's theorem about height then says that the relation

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

where v_c denotes the length of the line segment v_c etc. In other words, if we construct a square with side length v_c and a rectangle with side lengths c_a and c_b, then these figures would have the same content. Look at the following figure, in which just such a square and rectangle are highlighted:

Euclid's height theorem says that these figures have the same content. The red square has side length v_c, the green rectangle has one side length equal to c_a, and the other side length equal to c_b.

You can read the proof of Euclid's theorem about height on Wikipedia.

Euclid's theorem on the radius

This second theorem of Euclid also holds in a right triangle. Let's stay with the first picture of the triangle:

The second theorem then states that

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

If we construct a square with side length a, then this square will have the same content as a rectangle with side lengths c and c_a. Again, you can see the square and the rectangle:

Again, the proof can be found on Wikipedia.