Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

1. Given that $O$ is a point inside $\triangle A B C$ , and satisfies
$O A \cdot O B=O B \cdot O C=O C \cdot O A \text {. }$

Then point $O$ is the $\triangle A B C$ 's $\qquad$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

ニ、1. Orthocenter.
From $O A \cdot O B=O B \cdot O C$ we can get $O B \cdot A C=0$ , so, point $O$ lies on the altitude of side $A C$ . Similarly, we can prove that point $O$ also lies on the altitudes of the other sides. Therefore, point $O$ is the orthocenter of $\triangle A B C$ .

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