Pythagoras
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We will look at the Phythagorean Theorem and show how the hypotenuse of a right angle triangle on a flat plan can be determined from the other two sides of the triangle. This theorem is credited to Pythagoras although it is generally assumed that it was actually known before Pythagoras' name was attached to it.

In words this theorem states that for a right angle triangle the square of the hypotenuse equals to the sum of the squares of the other two sides of the triangle.

Proof by similar triangles

Let's start with the following triangle with a 90o angle between the vertical and horizontal sides. If we have the values of the vertical and horizontal sides (AB and BC) we would like to get the value of the other side (AC - opposite the 90o angle).
A B C
In order to accomplish this we can construct a line from the 900 angle (point B) to the opposite side (AC) in such a way that a 90o angle is formed at H on AC.
Θ Φ A Θ Φ B C H
From our basic geometry knowledge of proportions of triangles we know that ∠ABH (Θ) must be the same as ∠ACB and ∠CBH (Φ) must be the same as ∠CAB. Thus it means that triangle ABH, CBH and CAB have the same proportions and therefore we can make the following deductions:
$$ \begin{aligned} {{AH} \over {AB}} & = {{AB} \over {AC}}\\ {{CH} \over {BC}} & = {{BC} \over {AC}}\\ \therefore\, {{AH} \cdot {AC}} & = {AB}^2\\ {{CH} \cdot {AC}} & = {BC}^2\\ \therefore\, {AB}^2 + {BC}^2 & = {{AH} \cdot {AC}} + {{CH} \cdot {AC}}\\ & = {AC} \cdot \left( {AH} + {CH} \right)\\ & = {AC} \cdot {AC}\\ \therefore\, {AB}^2 + {BC}^2 & = {AC}^2\\ \end{aligned} $$