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. q.e.d.
