ABC is a triangle in which none of the angles is obtuse. The perpendicular AD from A to BC is produced to meet the circumcircle of the triangle at E. If D is equidistant from A and E prove that the triangle must be right-angled. If, alternatively, the incentre of the triangle is equidistant from A and E, prove that cos B + cos C = 1.
The Attempt at a Solution
I can see that D must lie on the diameter as must B and C, so A must be 90 degrees. The second part of the question is where I am stumped. I have been trying to find an expression that constrains the incircle to be equidistant from A and E and transform that expression to the required one. I toyed with r = (a+b+c) where r is the radius of the incircle and a, b, c are sides of the triangle. However, I couldn't make any progress.