Line, Circle, and Fixed Points

The applet below illustrates Problem 3 from the 2007 Irish Mathematical Olympiad:

The point P is a fixed point on a circle and Q is a fixed point on a line. The point R is a variable point on the circle such that P, Q and R are not collinear. The circle through P, Q and R meets the line again at V. Show that the line VR passes through a fixed point.

The applet actually illustrates a slightly different problem. Points R and V are interchangeable in the sense that one uniquely determines the other. As it was more convenient to keep V on a given line than P on a given circle, I followed an equivalent formulation of the problem:

The point P is a fixed point on a circle and Q is a fixed point on a line. The point V (different from Q) is a variable point on the line. The circle through P, Q and V meets the given circle again at R. Show that the line VR passes through a fixed point.

Explanation

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Three concurrent circles have several engaging properties, one of which plays a central role in the proof.

Given three circles concurrent in P. Let T be the point of intersection of the pair C(O) and C(PQV'). Then any chain of line segments VS (through R), SV' (through T), V'V (through Q) is closed.

(In the applet I actually picked V' at the intersection of SP with the given lines, making T = P and two circles C(O) and C(PQV') tangent - which in itself is not at all important.)

By the construction, V and V' are collinear with Q. By the above statement VR and V'P meet in point S on C(O). Since P and V' are fixed as is the given circle C(O), the point of intersection of PV' with C(O) is also fixed and does not depend on the position of V on line QV'!

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