Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/241

Rh PROPOSITION B. THEOREM.

If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle con-tained by the segments of the base, together with the square on the straight line which bisects the angle.

Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD: the rectangle BA, AC shall be equal to the rectangle BD,DC, together with the square on AD.

Describe the circle ACB about the triangle, [IV. 5. and produce AD to meet the circumference at E, and join EC. Then, because the angle BAD is equal to the angle EAC, [Hypothesis. and the angle ABD is equal to the angle ABC, for they are in the same segment of the circle, [III. 21. therefore the triangle BAD is equiangular to the triangle EAC. Therefore BA is to AD as BA is to AC; [VI. 4. therefore the rectangle BA, AC is equal to the rectangle EA,AD, [VI. 16. that is, to the rectangle BD, DA, together with the square on AD. [II. 3, But the rectangle JSD, DA is equal to the rectangle BD,DC; [III.35. therefore the rectangle BA, AC is equal to the rectangle BD, DC, together with the square on AD.

Wherefore, if the vertical angle &c.