Ralph Osborn, Midshipman at Annapolis: A Story of Life at the U.S. Naval Academy

by Edward L. Beach

Demonstration:

Let ABC be a right-angled triangle.

To prove (AB)² = (AC)² + (BC)².

On AB erect the square ABED. Draw BM parallel and equal to AC. From A erect a perpendicular through M.

By inspection triangle AMB [=] triangle ABC.

From point E drop a perpendicular to the line BM. The triangle thus formed, ENB, having a side and angles equal to a side and angles of triangle AMB, is seen by inspection to be equal to triangle AMB, and therefore to triangle ABC. Hence BN is equal to BC.

In a similar way construct the triangles ADH and DKE. By inspection each of these is seen to be equal to triangle ABC.

The square erected on AB is thus equal to four times the triangle ABC plus the rectangle HMNK.

Rectangle HMNK = MN × MH.

MN = BM - BN = AC - BC.