Is there a formal name for the point which is the reflection of the incenter about the circumcenter of a triangle?
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asked Jul 28, 2023 at 5:23
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1$\begingroup$ Try computing its trilinear or barycentric coordinates, then look the result up in the encyclopedia of triangle centers. $\endgroup$– Noah SchweberCommented Jul 28, 2023 at 14:14
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1 Answer
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The reflection of the incenter about the circumcenter is the Bevan point of the triangle. Quoting from the mathworld page:
The Bevan point $V$ of a triangle $\triangle ABC$ is the circumcenter of the excentral triangle $\triangle J_AJ_BJ_C$. It is named in honor of Benjamin Bevan, a relatively unknown Englishman [who] proposed the problem of proving that the circumcenter $O$ was the midpoint of the incenter $I$ and the circumcenter of the excentral triangle and that the circumradius of the excentral triangle was $2R$ (Bevan 1806), a problem solved by John Butterworth (1806).
