Consider acute-angled triangles $ ABC$ and $ APQ$, where $ P$ and $ Q$ lie on the side $ BC.$ Prove that the circumcenter of $ \triangle ABC$ is closer to line $ BC$ than the circumcenter of $ \triangle APQ.$
Source: Ukraine 1997
Tags: geometry, circumcircle, geometry unsolved
Consider acute-angled triangles $ ABC$ and $ APQ$, where $ P$ and $ Q$ lie on the side $ BC.$ Prove that the circumcenter of $ \triangle ABC$ is closer to line $ BC$ than the circumcenter of $ \triangle APQ.$