Question 21
Let $ABC$ be a triangle and $P$ be a point in its plane inside the region of the angle $BAC$, but lying outside triangle $ABC$.
a. Prove that any two of the following statements imply the third:
(i) the circumcentre of triangle $BPC$ lies on the ray $\vec{PA}$;
(ii) the circumcentre of triangle $CPA$ lies on the ray $\vec{PB}$;
(iii) the circumcentre of triangle $APB$ lies on the ray $\vec{PC}$.
b. Prove that when the conditions in (a) hold, the circumcentres of triangles $BPC$, $CPA$, $APB$ lie on the circumcircle of triangle $ABC$.
a. Prove that any two of the following statements imply the third:
(i) the circumcentre of triangle $BPC$ lies on the ray $\vec{PA}$;
(ii) the circumcentre of triangle $CPA$ lies on the ray $\vec{PB}$;
(iii) the circumcentre of triangle $APB$ lies on the ray $\vec{PC}$.
b. Prove that when the conditions in (a) hold, the circumcentres of triangles $BPC$, $CPA$, $APB$ lie on the circumcircle of triangle $ABC$.