Let $A$ and $B$ be two fixed points on a fixed straight line. Two circles touch this line at $A$ and $B$ respectively and tangent to each other at $M$. When the circles vary, what does the locus of $M$ form?
2026-04-24 03:34:23.1777001663
What does the locus of $M$ form?
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It is a circle. Observe that each of the two circles is perpendicular to the circle $C$ with diameter $AB$, because each is perpendicular to $C$ at $A$ or $B$ respectively. This means that for them to be tangent they have to intersect $C$ at the same point besides $A$ or $B$ respectively. Hence $M$ is on $C$, and clearly all points on $C$ are achievable if we allow degenerate circles with radius $0$ or $\infty$.