
Orthocentres
1. An acute triangle PQR has circumcentre o and orthocentre h. Extend RH to meet PQ at S. The line perpendicular to OS through S meets RP at T. Show that $\displaystyle \angle SHT = \angle QPR. $
2. ABCD is a cyclic quadrilateral, and midpoints are drawn on every side of the cyclic quad. Show that by linking the orthocentre of the 4 "corner" triangles, we get a parallelogram. (A "Corner Triangle" is made by joining the midpoints of 2 adjacent sides of the cyclic quad., along with one corner as a vertice).