Perfect Triangles

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

x = s - a
y = s - b
z = s - c


s = x + y + z

a = z + y

b = x + z

c = x + y


x + y + z = (s - a) + (s - b) + (s - c) = 3s - (a + b + c) = 3s - 2s = s

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 .  .  .