Related to Euclid, Elements VI.31

The Altitude's Shadow

Traditional · Antiquity · Similar trianglesDifficultyClassical

In a right triangle, the square on the hypotenuse equals the sum of the squares on the two legs — provable entirely from the proportions a single altitude creates, without ever measuring an area.

Let triangle $A B C$ be right-angled at $A$ . Whatever else is true, three squares — on $A B$ , on $A C$ , on $B C$ — sit waiting on its sides.
Drop a perpendicular from $A$ to the hypotenuse, meeting it at $D$ . This one stroke cuts triangle $A B C$ into two smaller triangles, $A B D$ and $A C D$ .
Triangle $A B D$ shares angle $B$ with triangle $A B C$ , and both have a right angle. Two matching angles make them similar.
By the same reasoning — sharing angle $C$ , both right-angled — triangle $A C D$ is similar to triangle $A B C$ as well.
Similar triangles keep their sides in proportion. From $A B D \sim A B C$ : $A B^{2} = B C \cdot B D$ . From $A C D \sim A B C$ : $A C^{2} = B C \cdot C D$ .
Add the two. Since $B D + C D$ is the whole of $B C$ : $A B^{2} + A C^{2} = B C \cdot (B D + C D) = B C^{2}$ .

Therefore c² = a² + b². ∎

The same altitude also proves the geometric-mean relation $AD^2 = BD \cdot CD$ — a fact usually taught right alongside this proof, and provable by the identical similar-triangle argument.