Question 42
Equilateral triangles $\triangle ABD$, $\triangle ACE$, and $\triangle BCF$ are drawn outside $\triangle ABC$ on each of its sides, with centers $G$, $H$, and $I$, respectively.
a) Prove that $\triangle GHI$ is equilateral.
b) Let $G'$, $H'$, and $I'$ be the reflections of $G$, $H$, and $I$ across $AB$, $BC$, and $CA$, respectively. Prove that $\triangle GHI$ and $\triangle G'H'I'$ have the same circumcenter.
a) Prove that $\triangle GHI$ is equilateral.
b) Let $G'$, $H'$, and $I'$ be the reflections of $G$, $H$, and $I$ across $AB$, $BC$, and $CA$, respectively. Prove that $\triangle GHI$ and $\triangle G'H'I'$ have the same circumcenter.