In the world of triangles with sides of integral length, the Heron triangle has an integral area. Heron triangles are named after Heron of Alexandria, a Greek scientist who lived during the first century A.D. and is credited with the ingenious formula relating the area of a triangle to the length of its sides: .
Here a, b, c stand for the length of the sides and s = (a + b + c) / 2. Note that s stands for half of the perimeter, or the semi-perimeter. Primitive Heron triangles have side measurements whose greatest common divisor is one.
- List all the different primitive Heron triangles with area less than or equal to 100.
- How many different primitive Heron triangles are there with sides less than or equal to 25? 50? 100?
- Which of the triangles in questions 1 and 2 is closest to be an equilateral?
- Comment on the truth of the following statements:
- The length of a side of a Heron triangle is even.
- The length of a side of a Heron triangle is a multiple of 3.
- The length of a side of a Heron triangle is a multiple of 5.
- The area of a Heron triangle is a multiple of 6.
(Reprinted with permission from Exploratory Problems in Mathematics, copyright 1992 by the National Council of Teachers of Mathematics. All rights reserved.)