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Halving the Perimeter of a Heart-Shaped Curve

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A curve consists of two semicircles of radius 1 and one semicircle of radius 2. A chord through the origin and inclination divides the curve into two parts, red and blue. Which curve is longer?

Contributed by: Izidor Hafner (October 2013)
Open content licensed under CC BY-NC-SA

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The two parts have equal lengths. The blue part consists of a semicircle of length : arc of length and arc of length Together, these are . The length of the whole curve is . So the dashed line halves the perimeter.

Reference

[1] D. Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin Books, 1991, p. 30.

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