Circle Intersecting Circles

Geometry Level pending

Let $\omega_1, \omega_2$ be distinct fixed circles on a flat plane. A third circle $\omega_3$ in the same plane intersects $\omega_1$ at exactly two distinct points $A, B$ and $\omega_2$ at exactly two distinct points $C, D.$ Lines $AB$ and $CD$ intersect at point $P.$ Over all possible $\omega_3,$ what is the locus of $P?$

The situation described is impossible A pair of parallel lines A point A circle A line It depends on the placement of

\omega_1

and

\omega_2

1 solution

Steven Yuan
Jul 11, 2017

Note that $ABCD$ is cyclic, so $(PA)(PB) = (PC)(PD)$ by Power of a Point. However, $(PA)(PB)$ is the power of point $P$ with respect to $\omega_1,$ while $(PC)(PD)$ is the power of $P$ with respect to $\omega_2.$ This implies that the locus of $P$ is the set of all points $X$ such that the powers of $X$ with respect to $\omega_1$ and to $\omega_2$ are the same. This is just the radical axis of $\omega_1$ and $\omega_2,$ which we know to be a straight line. Thus, the locus of $P$ is a line .

Circle Intersecting Circles

1 solution

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