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Geometry Level pending

Let ABC be a triangle and I its incenter. Let Γ be the circle tangent to sides AB, AC, as well as the circumcircle of ABC. Let Γ touch AB and AC at X and Y , respectively. Then I is the what of XY ?

perpendicular bisector none trisector midpoint

Aayush Srivastava by Aayush srivastava , 21, India

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2 solutions

Caleb Townsend
Feb 5, 2015

Process of elimination: assuming the answer isn't "none," I I can only be the midpoint of X Y XY since I I is a point and the other 2 options are lines, not points.

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I is the midpoint of XY .Proof. Let the point of tangency between the two circles be T. Extend T X and T Y to meet the circumcircle of ABC again at P and Q respectively. Note that P and Q are the midpoint of the arcs AB and AC. Apply Pascal’s theorem to BACP T Q and we see that X, I, Y are collinear. Since I lies on the angle bisector of ∠XAY and AX = AY , I must be the midpoint of XY .

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