# A Property of Isogonal Lines

### Problem

Given angle $BAC,$ line $AP,$ and its reflection in the bisector of $\angle BAC$, say $AP'.$ Let $P_b,$ $P_c$ be the projections of $P$ on the rays $AC$ and $AB,$ respectively.

Then $P_bP_c \perp AP'.$

### Solution

The proof is simple.

Let $D$ be the intersection of $AP'$ and $P_bP_c.$ Then quadrilateral $AP_{b}PP_{c}$ is cyclic (and $AP$ is a diameter of its circumcircle.) Now, $\angle PP_{b}P_{c} = \angle PAP_{c}$ as two inscribed angles subtended by the same arc. It is also given that $\angle PAP_{c} = \angle DAP_{b},$ implying $\angle PP_{b}P_{c} = \angle DAP_{b}.$ The latter two angles have one pair of the sides perpendicular, therefore the same holds for the other pair of their sides.

### References

1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, p. 65
2. R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960, p. 156