C is a point on the semicircle with diameter AB. D is a point on the arc BC. M, P, N are the midpoints of AC, CD and BD. The circumcenters of ACP and BDP are O, O'. Show that MN and OO' are parallel.
Solution
Let the center of the circle be X and the radius r. Let ∠AXM = θ, ∠BXN = φ. Note that O is the intersection of XM and the perpendicular to CD at Q, the midpoint of CP. We have XM = r cos θ. Let CD and XM meet at Y. Then ∠PYX = 90o - ∠PXY = 90o - ∠PXC - ∠CXM = θ + φ - φ = θ. Hence OX = PQ sec φ, so OX/XM = PQ/(r cos θ cos φ). Similarly, O'X/ON = PQ/(r cos θ cos φ), so OO' and MN are parallel.
© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 04
Last corrected 1 Jan 04