Reply To: Maths
Here are two geometry problems I have written; the first is definitely easier than the second!
1) Let ABC be an acute triangle. Let M be the midpoint of BC. Let D be the second intersection of AM with the circumcircle of ABC and let X be the reflection of D over M. Let E and F be the second intersections respectively of BX and CX with the circumcircle of ABC. Prove that AE*AF = AX^2.
2) Let ABC be a triangle with incenter I and incircle \omega. Let P, Q, R be the intersections of \omega with AI, BI, CI. Let l_A, l_B, l_C be the tangents to \omega through P, Q, R. Let X be the intersection of l_B and l_C, and define Y and Z similarly. Prove that AX, BY, CZ are concurrent.
I haven’t checked these and I usually ask for solutions up front, but since we are rather busy with Challenge 10 I thought I would throw them out there to see if the author gets any bites. Harry