# 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.
## Links

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

Back to my home page

*Last Updated Sep 07 2022*

Michael Somos <michael.somos@gmail.com>

Michael Somos
`"https://grail.eecs.csuohio.edu/~somos/"`