Perfect Triangles are triangles with integer sides, integer area, and numerically equal area and perimeter. Find all such triangles.
Our triangle with sides of lengths a, b , and c has area and permeter numerically equal. That is
and so when the area and perimeter are numerically equal,
Searching for integer values of a, b, and c can be simpler by substituting
s = x + y + z
a = z + y
b = x + z
c = x + y
and the relation with numerically equal perimeter and area becomes
4(x + y + z) = xyz
There is no loss of generality to assume that . Note that this means x and y will be the lengths adjacent to the largest angle -- opposite the longest side.
Can you find integer x, y, and z to satisfy the equation 4(x + y + z) = xyz ?
One solution, after you have tried . . .