The figure below shows a triangle ABC. A
line that passes through the orthocenter H intersects AB, BC, and CA
extended at D, E, and F, respectively. A_{1} is the reflection of D on BC,
B_{1} is the reflection of E on AC, and C_{1} is the reflection of E on AB.
Prove that (1) A_{1}E, B_{1}F, and C_{1}D are concurrent at P. (2) P is on the
circumcircle O.
