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A Heron triangle is a triangle with integral sides and integral area. For example, the familiar right-angled triangle (3,4,5) is a Heron triangle. Two Heron polygons are said to be Heron equivalent if they have the same perimeter and area. One natural question here is that if two triangles are Heron equivalent, should they necessarily be congruent? It is very well-known that the answer is in the negative. In this talk we show the existence of infinitely many Heron triangles which are Heron equivalent but not congruent. Also we show that no right-angled Heron triangle will be Heron equivalent to a Heron rectangle. However, there are infinitely many isosceles triangles which are Heron equivalent to rectangles. One technique for proving such theorems is by associating elliptic curves to triangles and use the powerful algebro-geometric structure available on these curves to prove the desired results.

I thank Dr. P. N. Shivakumar for mentioning this "Heron equivalence =?=> Congruence" problem.

1. Bremner, Andrew. On Heron triangles, Ann. Math, Inform, 33 (2006), 15-21.
2. Bremner, Andrew; Guy, Richard K. Triangle pairs with a common area and a common perimeter, Int. J. Number Theory 2 (2006) 217-223.
3. Goins, Edray; Maddox, Davin. Heron triangles via elliptic curves, Rocky Mountain J. Math. 36 (2006) 1511-1526.