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ratio of perimeters = ratio of circles

Source: Polish MO second round 1957 p2

August 29, 2024
ratiogeometryperimeterequal ratio

Problem Statement

Prove that if M M , N N , P P are the feet of the altitudes of acute-angled triangle ABC ABC , then the ratio of the perimeter of triangle MNP MNP to the perimeter of triangle ABC ABC is equal to the ratio of the radius of the circle inscribed in triangle ABC ABC to the radius of the circle circumscribed about triangle ABC ABC .
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