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ELLIPTIC CURVES COMING FROM HERON TRIANGLES

By Andrej Dujella, Juan and Carlos Peral

Abstract

Abstract. Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quartic v 2 = u(u − a)(u − b)(u − c). The Heron formula states that Q = √ P (P − a)(P − b)(P − c) where P is the semi-perimeter of the triangle, so the point (u, v) = (P, Q) is a rational point on the quartic. Also the point of in nity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curve y 2 = (x + a b)(x + b c)(x + c a). The point (P, Q) on the quartic transforms to −2abc (x, y)

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.306.3787
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