Heron Angles, Heron Triangles, and Heron Parallelograms

TL;DR

This paper introduces bijective parametrizations for Heron angles, Heron triangles, and Heron parallelograms, characterizing their rational side lengths, diagonals, and areas, thus advancing the understanding of these rational geometric figures.

Contribution

It provides the first explicit bijective parametrizations linking Heron angles, Heron triangles, and Heron parallelograms, unifying their rational properties.

Findings

Bijective parametrizations for Heron angles, triangles, and parallelograms.

Complete characterization of rational sides, diagonals, and areas.

Unified framework for Heron geometric figures.

Abstract

Heron angle: both its sine and cosine are rational Heron triangle: all its sides and area are rational Heron Parallelogram: all its sides, diagonals and area are rational We give one-to-one (bijective) parametrizations for all three concepts.

Equations

$$x^2+2axy+y^2=z^2$$

$$x$$

$$y$$

$$z$$

$$\text{conversely}2\lambda$$

$$x^2+(y+ax{)}^2=z^2+(ax{)}^2$$

$$y+ax=s_1+{\lambda }_2{s}_{2}ax=s_1-{\lambda }_2{s}_2$$

$$x=s_2-{\lambda }_2{s}_{1}z=s_2+{\lambda }_2{s}_1$$

$$2s_1$$

$$2s_2$$

$$2{\lambda }_2{s}_2$$

$$ax=a_1(s_2-{\lambda }_2{s}_1)=s_1-{\lambda }_2{s}_2$$

$$s_1=\frac{a+{\lambda }_2}{1+a{\lambda }_2}{s}_2$$

$$\begin{array}{rl}x& =s_2-{\lambda }_2\frac{a+{\lambda }_2}{1+a{\lambda }_2}{s}_2\\ x& =\frac{s_2}{1+a{\lambda }_2}(1-{\lambda }_2^2)\end{array}$$

$$\begin{array}{rl}z& =s_2+{\lambda }_2\frac{a+{\lambda }_2}{1+a{\lambda }_2}{s}_2\\ z& =\frac{s_2}{1+a{\lambda }_2}(1+2a{\lambda }_2+{\lambda }_2^2)\end{array}$$

$$y$$

$$m^2+\epsilon mn+n^2=1$$

$$m$$

$$n$$

$$\sigma$$

$${\lambda }^2=\frac{m}{n}\frac{1-n^2}{1+m^2}$$

$$\lambda =\frac{m}{n}$$

$$m(1-m^2)$$

$$m-n$$

$$m^2+mn+n^2=1$$

$$m=\frac{1-{\sigma }^2}{1+\sigma +{\sigma }^2},n=\frac{\sigma (2+\sigma )}{1+\sigma +{\sigma }^2},\lambda =\frac{1-{\sigma }^2}{\sigma (2+\sigma )}$$

$$\sigma =\frac{n}{1+m}$$

$$\lambda =mn$$

$$m{n}^3(1-m^2)$$

$$1+m^3{n}^3$$

$$1-mn+m^2{n}^2=n^2$$

$$\bigg{(}\frac{1}{n}\bigg{)}^{2}-m\bigg{(}\frac{1}{n}\bigg{)}+m^{2}=1$$

$$m=\frac{1-{\sigma }^2}{1-\sigma +{\sigma }^2},n=\frac{1-\sigma +{\sigma }^2}{\sigma (2-\sigma )},\lambda =\frac{1-{\sigma }^2}{\sigma (2-\sigma )}$$

$$\sigma =\frac{1}{n(1+m)}$$

$$\sigma =\frac{1}{2},m=\frac{3}{7},n=\frac{5}{7},\lambda =\frac{3}{5}$$

Full text

Heron triangle: all its sides and area are rational

Heron Parallelogram: all its sides, diagonals and area are rational

We give one-to-one (bijective) parametrizations for all three concepts.

1 Introduction

Special angles, triangles, rectangles, and parallelograms have been studied for ages by Mathematician and Scientists (Pythagoras, Heron, Diophantus, Brahmagupta to mention a few). Parameter representations, some in integers, have been found in all cases [1]. However, there was a lack in retrieving the parameters. Here we give parameter representations that are bijective (one-to-one).

2 Bijective parameter representation of a relation

Theorem 1

let $a,x,y,z$ be real numbers. Given $a$, the relation

[TABLE]

has the following bijective parameter representation, with parameters $\lambda,\sigma$

[TABLE]

Proof.

The relation (1) also reads

[TABLE]

∎

According to [2] this relation has the following bijective parameter representation with parameters $s_{1},s_{2},\lambda_{2}$

[TABLE]

Conversely

[TABLE]

Now

[TABLE]

resulting in

[TABLE]

Then

[TABLE]

[TABLE]

and

[TABLE]

Finally let $s_{2}=\lambda,\hskip 4.26773pt\lambda_{2}=\sigma$

Corollary 1

The relation, given $\varepsilon$,

[TABLE]

has the following bijective parameter representation, with parameter $\sigma$

[TABLE]

Conversely

[TABLE]

Proof.

In (2, 3, 4, 5) let $x=m,\hskip 4.26773pty=n,\hskip 4.26773ptz=1,$ and $2a=\varepsilon$ ∎

3 The relation

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We have two one-parameter families of fundamental solutions given by

1. (a)

Type I:

[TABLE]

Then

[TABLE]

or

[TABLE]

This relation has already been considered by Diophantus of Alexandria [3]

According to (17), the relation (23) has the bijective parameter representation with parameter $\sigma$, as

[TABLE]

conversely

[TABLE] 2. (b)

Type II:

[TABLE]

Then

[TABLE]

or

[TABLE]

and then

[TABLE]

According to (17), the relation (27) has the bijective parameter representation with parameter $\sigma$, as

[TABLE]

conversely

[TABLE]

Example

1. (a)

[TABLE] 2. (b)

[TABLE]

4 Heron angles

Definition 1

An angle $\alpha$ is called a Heron angle if both $\sin\alpha$ and $\cos\alpha$ are rational 2. 2.

The generator of an angle $\alpha$ is defined by

[TABLE]

Lemma 1

For $0<\alpha<\pi,\hskip 5.69046ptm(\alpha)>0$ and is an increasing function of $\alpha$.

Proof.

Since $\sin\alpha>0$ and $-1<\cos\alpha<1$ we see from (32) that $m(\alpha)>0$.

From

[TABLE]

we see that $m(\alpha)$ is increasing. ∎

Lemma 2

Given the generator $m=m(\alpha),\hskip 5.69046pt0<\alpha<\pi$, we find

[TABLE]

Proof.

From

[TABLE]

we find

[TABLE]

Observe that the generator of a Heron angle is rational and vice versa. Thus there is a one-to-one relationship between rational numbers and Heron angles.

∎

5 Heron triangles

For a triangle with sides $u_{1},u_{2},u_{3}$ and interior angles $\phi_{1},\phi_{2},\phi_{3}$, where $\phi_{k}$ is the angle opposite $u_{k}$, $k=1,2,3$

We have

The law of sine

[TABLE] 2. 2.

The law of cosine

[TABLE]

with similar relations involving the other sides and angles 3. 3.

The area $A$ of the triangle is given by

[TABLE]

and similarly for the other sides and angles.

Definition 2

A triangle is called a Heron triangle if all its sides and area are rational.

Observe that from (35, 36) all the interior angles are Heron angles.

For a Heron triangle we find from (34) and $\phi_{1}=\pi-(\phi_{2}+\phi_{3})$ the representation

[TABLE]

where $w$ is a rational scaling parameter.

Let now $q$ be the generator of $\phi_{2}$, $p$ the generator of $\phi_{3}$, i.e.

[TABLE]

Then

[TABLE]

[TABLE]

Introduce the new rational scaling parameter $v$ by

[TABLE]

Then we have the representation, with $p>0,q>0,v>0$ and $pq<1$

[TABLE]

This representation has the three parameters $p,q,v$. We now can retrieve these parameters from $u_{1},u_{2},u_{3},A$ as follows:

From

[TABLE]

we find the generator

[TABLE]

and similarly

[TABLE]

The scaling factor $v$ is then given by

[TABLE]

This is our one-to-one relationship.

Example

[TABLE]

6 Heron parallelograms

A parallelogram has sides $u_{1},u_{2}$ and diagonals $u_{3},u_{4}$.

Definition 3

1. (a)

A parallelogram with its sides and diagonals being rational is called a rational parallelogram. 2. (b)

A rational parallelogram with rational area is called a Heron parallelogram.

In [4] we found a bijective parameter representation for rational parallelograms.

Parameters $u>0,\hskip 7.11317pt0<m<1,\hskip 7.11317pt0<n<1$ and all rational

[TABLE]

conversly

[TABLE]

Not let $\phi$ be then angle between the sides $u_{1},u_{2}$. According to the law of cosine we find

[TABLE]

From the parallelogram equation

[TABLE]

we get

[TABLE]

[TABLE]

Now the area $A$ of a parallelogram is given by

[TABLE]

where

[TABLE]

[TABLE]

or according to (21) with

[TABLE]

we have

[TABLE]

The generator $p$ of the angle $\phi$ is given by

[TABLE]

where

[TABLE]

[TABLE]

Therefore

[TABLE]

The area is now given by

[TABLE]

Now, a Heron parallelogram with sides $u_{1},u_{2}$, diagonals $u_{3},u_{4}$ and area $A$ is parameterized by

[TABLE]

For a type I Heron parallelogram, we have the relation (22) and thus the bijective parameter $\sigma$. From (24, 25, 70) we find

[TABLE]

For a type II Heron parallelogram, we have the relation (26) and thus the bijective parameter $\sigma$. From (28, 29, 70) we find

[TABLE]

Observe that Heron parallelograms cover the case of Heron triangles with a rational median.

Finally, we have the relations

[TABLE]

Example

Type I : From (30)

[TABLE]

Type II : From (31)

[TABLE]