## The 2018 South African Mathematics Olympiad — Problem 4

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the week leading up to the contest, I plan to take a look at some of the problems from the senior paper from 2018. A list of all of the posts can be found here.

Today we will look at the fourth problem from the 2018 South African Mathematics Olympiad:

Let be a triangle with circumradius , and let be the altitudes through respectively. The altitudes meet at . Let be an arbitrary point in the same plane as . The feet of the perpendicular lines through onto are respectively. Prove that the areas of and satisfy the following equation:

Once again, we begin by creating a diagram. Again, since I already know how the solution plays out, I’ve drawn in the circle that passes through , and . We do know yet that these points are concylic, however, as it is not given directly in the problem statement.…