O is the circumcenter of the triangle ABC. XY is the diameter of the circumcircle perpendicular to BC. It meets BC at M. X is closer to M than Y. Z is the point on MY such that MZ = MX. W is the midpoint of AZ. Show that W lies on the circle through the midpoints of the sides of ABC. Show that MW is perpendicular to AY.
Solution
XW and AM are medians of AZX, so they meet at the centroid G of AZX and AG = 2/3 AM. But AM is also a median of ABC, so G is also the centroid of ABC. Now consider the similarity, center G, which takes X to W. It takes A, B, C to the midpoints of the opposite sides and hence the circumcircle of ABC to the circle through the midpoints. But X lies on the circumcircle of ABC, so W lies on the circle through the midpoints.
The similarity takes MW to AX, but the similarity takes lines to lines which are parallel. AX is perpendicular to AY (because XY is a diameter of the circumcircle through AXY). Hence MW is also perpendicular to AY.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002
Last corrected/updated 28 Apr 03