Pay attention to what happens when a line is drawn directly from the center of a right angle to its subtended line in a right-angled triangle: it divides itself towards its hypotenuse, creating two additional right angles within each side. Coincidentally, these two new figures have common features of each other as well as of the original or given triangle. This leads to Euclid's mathematical assertion that ˜if in a right-angled triangle a perpendicular be drawn from the right angle to the base, the triangles adjoining the perpendicular are similar both to the whole and to one another' [Proposition 8, Book 6]. .
Similarity between figures enables one to make comparisons utilizing proportionality. The following are pertinent propositions and an important definition which enhance this theorem. Proposition 32, Book 1: In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles and the three interior angles of the triangle are equal to two right angles. (180 degrees) Proposition 4, Book 6: In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which subtend the equal angles. Book 6, Definition 1: Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.
The Proof .
From the given triangle ABC which has a right angle of BAC, a perpendicular AD is drawn from point A to point B. Now because angle A is a right angle like angle BDA, and the angle B is common to both triangles and the remaining angle C equals angle DAB, one can infer that triangles ABC and DBA are equiangular. Because each triangle is equiangular I can justify that the sides are proportional as well. Thus, in comparing the triangle DAC in a like manner to triangle BDA, I can conclude that these triangles are similar to each other as well as to the whole ABC.