### Restore ABC if A, P, W are given

AoPS

The bisector of angle *A* of triangle *ABC* meet its circumcircle ω at point *W*. The circle 𝓈 with diameter *AH* (*H* is the orthocenter of *ABC*) meets ω for the second time at point *P.* Restore the triangle *ABC* if the points *A, P, W* are given.

We have to *AP*⊥*PH*; Furthermore, *PH* cuts *BC* at its midpoint. Taking this into account, the construction of triangle *ABC*, given *A, P, W*, is immediate:

• The center *O* of the circle (*APW*) is constructed, which must be the circumcenter of *ABC*.

• The perpendicular to *AP* through *P* cuts *OW* at *M* (midpoint of *BC*).

• The points of intersection of the circle (*APW*) with the perpendicular to *OW* through *M* are the vertices *B* and *C* of the triangle to be constructed.

NOTE: Once the points *A* and *P* are fixed, the construction of triangle *ABC* is NOT possible if *W* is located in the region between the perpendiculars to *AP* in *A* and *P*.

Hechos Geométricos en el Triángulo

Angel Montesdeoca. Dic. 2023