Perfect Trianglesare triangles with integer sides, integer area, and numerically equal area and perimeter. Find all such triangles.

## Analysis

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,

or

Searching for integer values of a, b, and c can be simpler by substituting

y = s - b

z = s - c

Note:

s = x + y + z

a = z + y

b = x + z

c = x + y

Then,

and the relation with numerically equal perimeter and area becomes

4(x + y + z) = xyzThere 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 . . .