Perfect Triangles

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.

Observation. If we relax the requirement that we have integer values, triangles with numerically equal perimeter and area will have an incircle with radius equal to 2. To see this consider this figure:

The area of the triangle can be seen as the sum of the areas of three triangular pieces with a common vertex at the incenter of the triangle. Thus

But if the perimeter is numerically equal to the area, then

and so, r = 2. Conversely, if the incircle of a triangle is 2, then the area is

which is the perimeter.

a. What right triangles with integer sides are Perfect triangles?

b. Find other (all?) Perfect triangles.

Hint. Consider using Heron's formula

Suggestion. Will the shortest side of a perfect triangle be longer than 4? Why? Try using some GSP constructions to explore this.

Suggestion. Will the shortest side be as much as 8 (twice the radius of the incircle)?

Suggestion. Can a spreadsheet be set up to search for such triangles?

ANALYSIS. An analysis is presented using Heron's formula for the area of a triangle and algebra. Some algebra can also be used to narrow the search in the use of a spreadsheet.