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