A Locus for Nagel Points and Incircles

Geometry Level 4

If $B$ and $C$ are fixed points at $B(-1, 0)$ and $C(1, 0)$ , and $A$ is such that the Nagel Point of $\triangle ABC$ lies on its incircle and such that $BC$ is the shortest side of $\triangle ABC$ , then the area enclosed by the locus of points of $A$ is equal to $m\sqrt{n}\pi$ , where $m$ and $n$ are positive square-free integers. Find $m + n$ .

Inspiration

The answer is 8.

1 solution

Mark Hennings
Jan 20, 2021

As shown here , the perimeter of the triangle $ABC$ is four times its shortest side $BC$ , and hence is $8$ . Thus $AB + AC = 6$ for all possible points $A$ , and so the locus of $A$ is an ellipse with foci $B,C$ and semimajor axis $a=3$ . Thus $ae=1$ , so that $e = \tfrac13$ and hence the semiminor axis is $b=\sqrt{8}$ . Thus the area enclosed by the locus of $A$ is $\pi ab = 6\pi\sqrt{2}$ , and hence $m+n=\boxed{8}$ .

A Locus for Nagel Points and Incircles

The answer is 8.

1 solution

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