# Constructing a Lattice from a Heron Triangle

The triangle (15,34, 35; 252) is the smallest acute Heron triangle indecomposable into two Pythagorean triangles. Realize it as a lattice triangle, with one vertex at the origin.

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#### Solution Preview

Let the vertices of the triangle be given by (0, 0), (a, b), and (c, d).

We divide the work into cases, looking for a lattice triangle with sides of 15, 34, or 35, or proving that no such lattice triangle exists.

Case 1: the side length of 15 touches the origin.

Case 2: the side length of 15 does not touch the origin.

To analyze case 1, first assume that the side of length 15 lies between the origin and (a, b). Also, without loss of generality, assume that a and ...

#### Solution Summary

Constructing a lattice from a Heron Triangle is examined. Pythagorean triangles are determined.

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