In the same way ABC, ACD are equiangular and similar. Therefore also ABD, ACD are equiangular and similar. therefore BD: DA=AD: DC; BD.DC=AD2. COR. 2. In the similar triangles BCA, ACD, therefore Similarly therefore but therefore BC: CD+BD = BC2 : CA2 + AB2, or, the square on the hypothenuse is equal to the squares on the other two sides. COR. 3. Any figure on BC is equal to the similar and similarly described figures on BA and AC. For these figures are to one another as the squares on BC, BA, and AC. [IV. 9. THEOREM XIII. If on each of the sides of the right-angled triangle ABC, as diameter, a semicircle be described, the crescents ADBE, AFCG are together equal to the triangle ABC. For the semicircles CFA, ADB, and BAC are similar D E B figures described on the sides and the hypothenuse of the rightangled triangle BAC: Therefore the semicircles CFA and ADB are equal to the semicircle BAC. Take from each the segments BEA, AGC. [IV. 9. Therefore the crescents ADBE, AFCG are equal to the triangle ABC. This theorem possesses great historical interest, as the first example of the quadrature of a curvilinear figure, It is due to Hippocrates, 450 B.C. THEOREM XIV. If ABC be a triangle, and from B and C BE, CF be let fall perpendicular to CA, AB, or these produced, For the right-angled triangles ABE, ACF have a common or equal angle at A, and are therefore equiangular and |