Heronian triangle

The geometry is meant by a hero American triangle a triangle wherein the side length and the area are rational numbers. It is named after Heron of Alexandria. Each triangle whose side lengths form a Pythagorean triple is heronisch because the side lengths of a triangle are integers and such because its area is equal to half the product of the two shorter side lengths. (Namely from the reversal of the Pythagorean theorem follows the squareness of the triangle. )

However, a hero African triangle is not necessarily rectangular. This is demonstrated on the example of an isosceles triangle with side lengths 5, 5 and 6 This triangle can be calculated from two congruent right triangles with sides 3, 4, 5 put together. The area is therefore. The example can be easily generalized: Taking a Pythagorean triple (a, b, c ) where c is the greatest number and another Pythagorean triple ( a, d, e) with e as the largest number, so you can, as in the adjacent drawing recognizable, the corresponding triangles put together along the two sides of length a to a hero African triangle. The new triangle has side lengths c, e and b d for the surface area is obtained

It is now interesting to ask whether one receives each triangle African hero by this method, thus joining two right triangles that match in a Kathetenlänge. The answer is no. For example, the African hero triangle with side lengths of 0.5, 0.5 and 0.6, so the shrunken by a factor of 10 version of the triangle described above, of course, not be decomposed into sub-triangles with integer side lengths. The same is true for the African hero triangle with side lengths 5, 29, 30 and the area 72, as none of the three altitudes of this triangle is an integer. Allowed for Pythagorean triples but any rational (not necessary all ) numbers, so can the question be answered with yes. (Note that one can receive from each triplet rational numbers, characterized in that one divides the values ​​of a triplet of integers by the same integer. )

Set

Each hero African triangle can be divided into two right triangles whose side lengths are given by Pythagorean triples of rational numbers.

Proof of the theorem

Consider again the above diagram, this time it is assumed that C, E, B D, and the triangular area A are rational. We can assume that the names were chosen so that the page length is b d is the greatest. This ensures that the decision taken by the opposite corner of this page Lot is located inside the triangle. In order to demonstrate that the triple ( a, b ​​, c) and (a, d, e) are triple Pythagorean, one must prove that a, b and d are rational.

As for the triangular area

Applies, one can solve for a and thus finds

This calculation expression is rational, because all numbers of the right-hand side are rational. So it only remains to show that b and d are rational. From the Pythagorean theorem, applied to the two right triangles are obtained

And

Subtraction of these equations yields:

The right-hand side of the last equation has to be rational, since d are rational by hypothesis c, e and b. This proves that b - d is rational. From this statement it follows by the rationality of b d, that b and d are rational.