Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Prove it Soviet Union

Problem:

In the triangle $ABC$ , the length of the altitude from $A$ is not less than $BC$ , and the length of the altitude from $B$ is not less than $AC$ . Find the angles.

Solution

Solution:

Let $k$ be twice the area of the triangle. Then $k \geq BC^2$ , $k \geq AC^2$ and $k \leq AC \cdot BC$ , with equality in the last case only if $AC$ is perpendicular to $BC$ . Hence $AC$ and $BC$ have equal lengths and are perpendicular. So the angles are $90^\circ$ , $45^\circ$ , $45^\circ$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.