Department of Mathematics Education
Jim Wilson, EMT 725
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.
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
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.