Known values: $A = (x_{1}, y_{1})$, $B = (x_{2}, y_{2})$, and $r$ is the inradius of $\Delta ABC$. $\Delta ABC$ will always be scalene.
What are the coordinates of $C$? Do we have enough information to determine this?
2026-03-27 12:08:24.1774613304
Determining coordinates of a triangles vertex knowing the other two vertices and the triangles inradius
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Of course not. Take $I$ (the incenter) on a line whose distance from $AB$ is $r$, between the projections of $A$ and $B$ on such a line. The symmetric of $AB$ with respect to $AI$ and the symmetric of $AB$ with respect to $BI$ will meet at a candidate $C$-point. The locus of such $C$s is a hyperbola: