Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/243

Rh Let ABCD be any quadrilateral figure inscribed in a circle, and join AC,BD: the rectangle contained by AC, BD shall be equal to the two rectangles contained by AB,CD and AD,BC.

Make the angle ABE equal to the angle DBC; [I. 23. add to each of these equals the angle EBD, then the angle ABD is equal to the angle EBC. [Axiom 2. And the angle BDA is equal to the angle BCE, for they are in the same segment of the circle; [III.21. therefore the triangle ABD is equiangular to the triangle EBC. Therefore AD is to DB as EC is to CB; [VI. 4. therefore the rectangle AD, CB is equal to the rectangle DB, EC. [VI. 16.

Again, because the angle ABE is equal to the angle DBC, [Construction. and the angle BAE is equal to the angle BDC, for they are in the same segment of the circle; [III. 21. therefore the triangle ABE is equiangular to the triangle DBC. Therefore BA is to AE as BD is to DC; [VI. 4. therefore the rectangle BA, DC is equal to the rectangle AE, BD. [VI. 16.

But the rectangle AD, CB has been shewn equal to the rectangle DB, EC; therefore the rectangles AD, CB and BA, DC are together equal to the rectangles BD, EC and BD, AE; that is, to the rectangle BD, AC. [II. 1.

Wherefore, the rectangle contained &c.