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On lattice points which become vertices ofHeronian triangles *Koichi ARIMOTO 1 and Yasuyuki HIRANO2
1 The Joint Graduate School, Hyogo Universityof Teacher Education (Naruto University of Education)
2 Naruto University of Education

1Introduction

Heronian triangles are triangles whose side lengths and area are all integers.

For example, the triangles with side lengths 3, 4 and 5 (this triangle hasarea 6) , and with side lengths 5, 5 and 6 (this triangle has area 12), etc.
are Heronian triangles. Specially the triangle whose side lengths 3, 4 and 5
is Pythagorearb t,riar\iota(jle , that is, a right triangle whose side lengths are allintegers. We discuss the lattice points which become vertices of Heroniantriangles.
In section 2, we introduce classical results. We state on the existence of
circles containing a finite number of lattice points, and a finite number ofpoints whose distance between any pair of them is integer. Finally, we stateon the formula of Pythagorean triple.
In section 3, we state our main theorems. We claim that there exists nlattice points on a circle such that the distance between any pair of them isinteger. Next, using about the formula of Pythagorean triple, we show oneof the sets of lattice points which become vertices of Heronian triangles.

2

Classical results

2.1

rational points on a circle and lattice points on acircle

A point (x, y)\in \mathbb{R}^{2} is rational point if x and y are rational numbers, and apoint (x, y) is a lattice point if x and y are integers.
First, we address Pick’s Theorem. Consider a polygon P whose vertices lie
at lattice points. As mentioned above, Pick s Theorem allows us to determine*This paper is a preliminary version and a final version will be submitted to elsewhere.

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the area of P from the number of lattice points on the boundary of P, B(P) ,and the number of lattice points in the interior of P, I(P) . More specifically,
Pick’s Theorem states the following,Proposition 2.1 (Pick’s theorem, 1899)

Let P be a polygon in the planewith its vertices at lattice point. Then the area of P, A(P) , is given byA(P)= \frac{1}{2}B(P)+I(P)-1,where B(P) is the number of lattice points on the boundary of P and I(P)is the number of lattice points in the interior of P.
By this proposition, we see that the area of a polygon in the plane with its
vertices at lattice point is rational number. Now suppose that there exists
a regular triangle such that its vertices at lattice point, for a side length a,it has area \sqrt{3}a^{2}/4 , where a^{2} is a positive integer, hence this is a irrational
number, a contradiction. Therefore the regular triangle cannot be embeddedin the lattice. Containing in case other regular n ‐gon, we state the followingproposition.

Proposition 2.2
If n lattice points form a regular n ‐gon (n>2) , thenn=4 ; thus the square is the only regular polygon that can be embedded inthe lattice.

Pythagorean triangles are right triangles whose sides have integral lengths.
This proposition implies the following proposition.
Proposition 2.3

If \alpha is an acute angle of a Pythagorean triangle, thenumber \alpha/\pi is irrational number.

Proposition 2.4

The number of rational points on a circle is 0,1,2 , or\infty.

Corollary 2.5 For any integer n>0 , there exists at least one circle whichcontains more than n lattice points.

2.2

rational distance and integral distance

In Euclidean geometry, Ptolemy’s theorem is a relation between the foursides and two diagonals of a cyclic quadrilateral, that is, a quadrilateral
whose vertices lie on a common circle.

Proposition 2.6 (Ptolemy’s theorem)

If ABCD is a cyclic quadrilateral,

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thenAB \cdot CD+AD\cdot BC=AC\cdot BD.

This proposition implies the following proposition that was known to Euler.

Proposition 2.7 [1, Proposition 1.1],[2, Theorem 1.14]

There exist infinite

points on a circle such that the distance between any pair of them is rational
number.

Corollary 2.8 For any integer n>0 , there exist n points on a circle suchthat the distance between any pair of them is integer.

2.3

Pythagorean triples

A Pythagorean triple is a triple a, b, c of natural numbers that satisfy theequation a^{2}+b^{2}=c^{2} A triangle whose side lengths form a Pythagorean
triple is a Pythagorean triangle, and is necessarily a right triangle.
Proposition 2.9

Let a, b, c be a solution of the equationa^{2}+b^{2}=c^{2}in relatively prime integers. Then a or b must be even. On the assumptionthat a is even, a, b, c has the form

a=2uv, b=u^{2}-v^{2} , c=u^{2}+v^{2},
where u and v are relatively prime natural numbers with u+v odd and u>v.
Every such pair u, v corresponds to exactly one primitive Pythagorean triple
a, b, c , that i_{\mathcal{S} , greatest common divisor of a, b, c is equal to 1.

3

Main theorems

In connection with Proposition 2.4 and Proposition 2.7, we get the following

proposition. The basic idea of following proof is due to [2].
Proposition 3.1
There exist circles which contain infinite rational points
such that the distance between any pair of them is rational number.

Outline of the proof. Let a, b, c be a Pythagorean triple, now we consider acircumcircle \Gamma of isosceles triangle with side lengths c, c and 2a . Let \alpha be theangle subtended at the center by the chord with side length c , we consider
the circumcircle \Gamma with its center at the origin. Let P_{0} be a intersection
point of circumcircle I^{\urcorner} and y ‐axis, and we put P_{i}(i=1,2, \cdots) be points on

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the circumference equally spaced with central angle \alpha . The points P_{i}(i=0,1, \cdots) are all rational points and the distance between any pair of them isrational number(see Figure A).

\blacksquare

Lemma 3.2 If there exist three lattice points such that the distance between
any pair of them is integer, then the area of triangle with three lattice points
is integer, that is, they become vertices of Heronian triangle.
By these arguments, we get the following theorem.
Theorem 3.3
There exist circles which contain n lattice points \mathcal{S}uch thatthe distance between any pair of them is integer. Specially, in case n=3,
there are three lattice points on circles which become vertices of Heroniantriangles.

Next, using classical formula of Pythagorean triples, Proposition 2.9, we get
the following theorem.

Theorem 3.4 Not on the line three points which belong to the following set
consists of lattice points\{(0, \pm y), (0,0), (\pm x_{1},0), \cdot , (\pm x_{n}, 0)\}vertices of Heroman triangles (see Fvgure B ), where let p_{i}(i=, n) be odd prime number satisfying p_{i}>p_{j} (for i<j) , and p_{1}>be,come,1,2p_{2}\cdots p_{n} , we can write
y=4p_{1}\cdots p_{n}.
\cdot\cdot

Figure A

x_{k}=(p_{1}\cdots p_{k})^{2}-4(p_{k+1}\cdots p_{n})^{2}(k=1, \cdots , n) ,Figure B

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Acknowledgment
We would like to express their hearty thanks to the members in the RIMS
workshop for their valuable advices.

References[1] G.Campbell, Points on y=x^{2}

Computation, 73, 2093‐2108, 2004.

at Rational Distance, Mathematics of[2] P.Frankl, and H. Maehara, Kikagaku no Sanpomichi(in Japanese), Kyoritsu‐

Shuppan, 1991.

[3] J.Garbett, Lattice Point Geometry: Pvck’s Theorem and Minkowski’s The‐
orem, 2010, available at
http://documents.kenyon.edu/math/GarbettJSenEx2011.pdf[4] H.Hadwiger, and H.Debrunner,Holt, Rinehart and Winston, 1964.

Comb_{i} natorialGeomerty in the Plane,[5] H.Koch, Number Theory: Algebraic Numbers and Functions, Graduate

Studies in Mathematics, 24, American Mathematical Society, 2000.

Koichi ARIMOTOJOINT GRADUATE SCHOOL (PH.D. PROGRAM) IN SCIENCE OF SCHOOL EDUCA‐TION, HYOGO UNIVERSITY OF TEACHER EDUCATION(NARUTO UNIVERSITY OF EDUCATION),
KATO‐SHI, HYOGO 673‐1494, JAPANYasuyki HIRA NO
DEPARTMENT OF MATHEMATICS,NARUTO UNIVERSITY OF EDUCATION,NARUTO‐SHI, TOKUSHIMA 772‐8502, JAPAN