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All-Soviet-Union Mathematical Olympiad

Geometry Difficulty 4.4 AIME Prove it Soviet Union

Problem:

Prove that you can always draw a circle radius $A/P$ inside a convex polygon with area $A$ and perimeter $P$ .

Solution

Solution:

Draw a rectangle width $A/P$ on the inside of each side. The rectangles at each vertex must overlap since the angle at the vertex is less than $180$ . The total area of the rectangles is $A$ , so the area covered must be less than $A$ . Hence we can find a point not in any of the rectangles. But this point must be a distance more than $A/P$ from each side, so we can use it as the center of the required circle.

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