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AC=MC iff AM bisects angle DAB

Source: Romanian TST 2002

February 5, 2011
geometrygeometry proposed

Problem Statement

Let ABCABC be a triangle such that AC≠BC,AB<ACAC\not= BC,AB<AC and let KK be it's circumcircle. The tangent to KK at the point AA intersects the line BCBC at the point DD. Let K1K_1 be the circle tangent to KK and to the segments (AD),(BD)(AD),(BD). We denote by MM the point where K1K_1 touches (BD)(BD). Show that AC=MCAC=MC if and only if AMAM is the bisector of the ∠DAB\angle DAB.
Neculai Roman
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