An acute-angled triangle ABC is given, through the vertices B and C of which a circle Ω, A∈/Ω, is drawn. We consider all points P∈Ω, that do not lie on none of the lines AB and AC and for which the common tangents of the circumscribed circles of triangles APB and APC are not parallel. Let XP be the point of intersection of such two common tangents.
a) Prove that the locus of points XP lies to some two lines.
b) Prove that if the circle Ω passes through the orthocenter of the triangle ABC, then one of these lines is the line BC. geometryLocusUkrainian TYM