PROP. VIII.—Theorem.
The triangles (ACD, BCD) into which a right-angled triangle (ACB) is divided, by the perpendicular (CD) from the right angle (C) on the hypotenuse, are similar to the whole and to one another.
Dem.—Since the two triangles ADC, ACB have the angle A common, and the angles ADC, ACB equal, each being right, they are [I. xxxii.] equiangular; hence [iv.] they are similar. In like manner it may be proved that BDC is similar to ABC. Hence ADC, CDB are each similar to ACD, and therefore they are similar to one another.
Cor. 1.—The perpendicular CD is a mean proportional between the segments AD, DB of the hypotenuse.
For, since the triangles ADC, CDB are equiangular, we have AD : DC :: DC : DB; hence DC is a mean proportional between AD, DB (Def. iii.).
Cor. 2.—BC is a mean proportional between AB, BD; and AC between AB, AD.
Cor. 3.—The segments AD, DB are in the duplicate of AC : CB, or in other words, AD : DB :: AC2 : CB2,
Cor. 4.—BA : AD in the duplicate ratios of BA : AC; and AB : BD in the duplicate ratio of AB : BC.
PROP. IX.—Problem.
From a given right line (AB) to cut off any part required (i.e. to cut off any required submultiple)