Two Circles on a Side of a Triangle

Two Circles on a Side of a Triangle: What is this about?
A Mathematical Droodle

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The applet may suggest the following statement:

Let point P lie on a side AB of ΔABC. Circle C(A) passes through A and P and is tangent to AC, circle C(B) passes through B and P and is tangent to BC. Q is the second point of intersection of C(A) and C(B). Show that, regardless of the position of P on AB, line PQ passes through a fixed point X.

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Proof

First note that in C(A)

∠AQP = ∠CAP.

Similarly, in C(B),

∠BQP = ∠CBP.

If so, then

∠AQB + ∠ACB = 180°,

which says that quadrilateral ACBQ is cyclic so that Q lies on the circumcircle of ΔABC.

Let X be the point of intersection of PQ with the circumcircle. Inscribed angles AQX and BAC are equal. Therefore, the arcs AX and BC they subtend are also equal. It follows that CX is parallel to AB. Hence, X is independent of P.

Radical Axis and Radical Center

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