G2

Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC. Suppose
that ω is tangent to AB at B′ and to AC at C′. Suppose also that the circumcenter O of the
triangle ABC lies on the shorter arc B′C′ of ω. Prove that the circumcircle of ABC and ω
meet at two points.

Question

G2

Let ABC be an acute triangle. Let ω be a circle whose center L lies on the side BC. Suppose
that ω is tangent to AB at B′ and to AC at C′. Suppose also that the circumcenter O of the
triangle ABC lies on the shorter arc B′C′ of ω. Prove that the circumcircle of ABC and ω
meet at two points.

anonymous · Answer

Need to draw a picture to help me see what's going on....
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A little sloppy, but that's what I imagine it looks like.
Seems pretty tightly constrained, so should be able to proved by relating the various properties of the circles.

anonymous · Answer

Will have to use a lot of imagination to pretend that vertex A also touches the circumcircle, and that BC is a little straighter than I've drawn it, but again it shows some of the relative positions.

anonymous · Answer

I'm thinking that a key point is the distance between the two circles is equal to the radius of ω, and the diameter of ω is larger than the radius of the circumcircle. I'm going to have to draw a better picture to visualize it, but I think that's a step in the right direction.