Question 36
Let $f: \pi \to \mathbb{R}$ be a function from the Euclidean plane to the real numbers such that
$$
f(A) + f(B) + f(C) = f(O) + f(G) + f(H)
$$
for any acute triangle $ABC$ with circumcenter $O$, centroid $G$, and orthocenter $H$. Prove that $f$ is constant.
$$
f(A) + f(B) + f(C) = f(O) + f(G) + f(H)
$$
for any acute triangle $ABC$ with circumcenter $O$, centroid $G$, and orthocenter $H$. Prove that $f$ is constant.