Cor. 1.—If two triangles ACB, ADB on the same base AB, and on the same side of it, have equal vertical angles, the four points A, C, D, B are concyclic.
Cor. 2.—If A, B be two fixed points, and if C varies its position in such a way that the angle ACB retains the same value throughout, the locus of C is a circle.
In other words—Given the base of a triangle and the vertical angle, the locus of the vertex is a circle.
Exercises.
1. Given the base of a triangle and the vertical angle, find the locus—
(1) of the intersection of its perpendiculars;
(2) of the intersection of the internal bisectors of its base angles;
(3) of the intersection of the external bisectors of the base angles;
(4) of the intersection of the external bisector of one base angle and the internal bisector of the other.
2. If the sum of the squares of two lines be given, their sum is a maximum when the lines are equal.