Department of Mathematics Education
Jim Wilson, EMT 725

Perfect Triangles

Perfect Triangles are 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

or

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

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

Then,

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

Can you find integer x, y, and z to satisfy the equation 4(x + y + z) = xyz ? Return
One solution, continuing from above.
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