# Rational Triangles

## Definition

Define a Rational Triangle as a triangle in the Euclidean plane such that all three sides measured relative to each other are rational. Once, it was thought that all triangles were rational. The discovery of counterexamples is attributed to the Pythagoreans. Any triangle similar to a rational triangle is rational also. Take as a unit the greatest common measure of the three sides. Then the length of the sides are positive integers whose greatest common measure is unity. All rational triangles can be uniquely constructed in this way from three positive integers with greatest common divisor unity and each less than the sum of the other two (triangle inequality).

## Right Triangles

Define a Rational Right Triangle as a right triangle which is a rational triangle. A right triangle is a rational triangle if and only if all six trigonometrical ratios of the two complementary acute angles are rational. It can be proved, using the inscribed circle, that the tangent of its half angles are rational numbers. Define a Heronian Angle as an angle such that the tangent of its half is a rational number. Conversely, any rational number between zero and one is the tangent of half an acute angle of a rational right triangle. The rational number associated with one acute angle has numerator and denominator both odd, but for the other angle they are of different parity. All rational right triangles are constructed in this way from a rational number between zero and one in both ways.

## Heronian Triangles

Define a Heronian Triangle as a rational triangle with area rational relative to the square of any side. The name refers to the formula for the area of a triangle given the sides known as Hero's formula. This remarkable formula is one-fourth the square root of the product of four factors, one of which is the perimeter of the triangle (the sum of the three sides), and the other three factors are gotten by subtracting one side from the the sum of the other two sides. The simplest triangle is an equilateral triangle which is a rational triangle, but is not Heronian because the area is not rational relative to the square of a side. Next simplest triangle is an isoceles right triangle which has area one half the square of any leg, but the hypoteneuse is not rational relative to a leg as the Pyhtagoreans discovered, and so the triangle is not Heronian. The simplest example of a Heronian triangle is a right triangle with ratio of sides 3:4:5. In fact, any rational right triangle is a Heronian triangle. The area of any triangle is half the perimeter times radius of the inscribed circle. Thus, the inradius of a Heronian triangle is rational relative to the sides. A radius from the incenter to any side splits it into two segments of length the semiperimeter diminished by the non-adjacent side length. It follows that the tangent of any bisected angle of the triangle is a rational number and therefore the original angle is a Heronian angle. Thus, a Heronian triangle is the same as a triangle having three Heronian angles. However, it suffices to require two Heronian angles for the following reason. The area of any triangle is half of base times height. Thus, all the altitudes of a Heronian triangle are rational relative to the sides. Any altitude of a Heronian triangle splits the triangle into two Heronian right triangles. The original triangle is the sum or difference of the two right triangles depending on if the altitude is internal or external.

## Lattice Triangles

There is a formula for the area of a triangle computed as a quadratic function of the coordinates of its vertices. This formula is one-half of a sum of three terms, each of which is a difference of cross-products of the x and y coordinates of two points (a two by two determinant) in circular order. Thus, a triangle with rational coordinates has rational area, and, if the sides are rational, then the triangle is Heronian. Conversely, a Heronian triangle's vertices can be given rational coordinates. This is proven by an analysis of the reduction to the case of integer sides which leads to the triangle realization as an integer lattice triangle.

## Literature

Since the study of arithmetical properties of triangles goes back at least to Pythagoras the published literature is ancient and well worked. There is no universal terminology or notation so I have decided to use my own variation of commonly used terms. One useful source is a slim book Pythagorean Triangles by Waclaw Sierpinski. It was published in 1962 and Dover Publications reprinted it in 2003. Another source is chapter four of Mathematical Recreations by Maurice Kraitchik reprinted by Dover Publications in 1953. A recent journal article is "On basic Heronian triangles" by Kozhegeldinov in Mathematical Notes 55 (1994) pp. 151-156 [MR 95c:51016]. There are scattered articles in the American Mathematical Monthly. One recent article is in the February 1997 issue on "An Infinite Set of Heron Triangles with Two Rational Medians" by Ralph H. Buchholz and Randall L. Rathbun.

There is some discussion of rational triangles in the Geometry Junkyard. There is an entry for Heronian Triangle in MathWorld. The subject of triangles is extensive. Another angle on the subject is Clark Kimberling's Triangle Centers.

## Tables

Here is my Heronian Triangle Table

Here is my Pythagorean Triple Table