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This Demonstration illustrates Proposition 20 of Archimedes's work On Spirals.

Let

be any point on the first turn of the spiral, and let

be the intersection of the tangent to the spiral at

, with the perpendicular to

. Then

Reference

[1] T. L. Heath (ed.), The Works of Archimedes, New York: Dover Publications, 2002.

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Equality of a Segment and an Arc in Archimedes's Spiral