5. Any line intersecting the legs of a right angle is cut harmonically by any two lines through its vertex which make equal angles with either of its sides.
6. If the base of a triangle be given in magnitude and position, and the ratio of the sides, the locus of the vertex is a circle which divides the base harmonically in the ratio of the sides.
7. If a, b, c denote the sides of a triangle ABC, and D, D′ the points where the internal and external bisectors of A meet BC; prove
8. In the same case, if E, E′, F, F′ be points similarly determined on the sides CA, AB, respectively; prove
| + + = 0, | |||
| and | + + = 0. |
PROP. IV.—Theorem.
The sides about the equal angles of equiangular triangles (BAC, CDE) are proportional, and those which are opposite to the equal angles are homologous.
Dem.—Let the sides BC, CE, which are opposite to the equal angles A and D, be conceived to be placed so as to form one continuous line, the triangles being on the same side, and so that the equal angles BCA, CED may not have a common vertex.