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Problem 94

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Almost equilateral triangles

It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle $5-5-6$ has an area of $12$ square units.

We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than one unit.

Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area and whose perimeters do not exceed one billion $(1,000,000,000)$.

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几乎等边的三角形

容易证明，不存在边长和面积均为整数的等边三角形。但是，存在三边长$5-5-6$均为整数的几乎等边的三角形 ，其面积$12$同样为整数。

定义几乎等边的三角形为有两条边一样长且第三边长与这两边最多相差$1$的三角形。

考虑所有边长和面积均为整数、周长不超过十亿$(1,000,000,000)$、几乎等边的三角形，求其周长之和。

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