Project Euler #94 : Almost equilateral triangles

Here is the problem description:

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).

* update on 2019/1/6:
I dropped the original contents in compliance with Project Euler policy, discouraging posting solution or hint on public space. Message me if you would need help. I believe you should know Heron's formula.