Two Circles on Angle Bisector

The applet below is one of two attempts to illustrate a recent theorem by J. M. Unger. The other illustration is available elsewhere.

In ΔABC, (I) is the incircle and (P) is a circle tangent to AB and AC. A circle (O) passing through B, C, tangent to the circles (I) (internally) and (P) (externally), exists if and only if one of the intangents of (P) and (I) is parallel to BC.

(In the applet, the vertices of the triangle and point P on the bisector of ∠A are draggable. The applet displays two circles passing through B and C. One is tangent to (I) and the other to (P). To verify Unger's theorem drag P until the two circle coincide.)

It is worth noting that when one of the common internal tangents of (I) and (P) is parallel to BC, the other one is antiparallel to it. The sides AB and AC cut off each of the the coaxal circles through B and C (and, hence, the center on the perpendicular bisector of BC) a chord which also antiparallel to BC.

References

1. J. Marshall Unger, A new proof of a "hard but important" Sangaku problem, Forum Geometricorum, 10 (2010) 7--13.

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