Integer triangle with integer heights

In summary, "Integer triangle with integer heights" explores the properties of triangles whose vertices are at integer coordinates and whose altitudes from these vertices to the opposite sides are also integers. The study involves examining the relationships between the triangle's dimensions, such as side lengths and angles, and the integer constraints imposed on their heights. The research delves into specific cases, illustrating how these conditions can affect the classification and characteristics of the triangles involved.
  • #1
mathics
8
1
Homework Statement
Prove that if the lengths of the sides of a triangle and the heights drawn to all sides are integers, then these integers are all multiples.
Relevant Equations
Pythagorean theorem
I tried to figure it out by proof by contradiction. I assumed wlog that one side is a prime number, which gave a contradiction, and I also found that such a triangle should be isosceles. Then I assumed wlog that one of the heights is a prime number and I also got a contradiction. Therefore, all sides and heights are whole numbers, which are also multiples.

Is my idea for proofing okay? Is there a better way for proofing this?
 
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  • #3
PeroK said:
:welcome:

Are you talking about a Heronian triangle, which also has integer area?

https://en.wikipedia.org/wiki/Heronian_triangle
Yes. If the triangle has integer sides and integer heights (the heights also must multiples not primes), then I believe the area should also be integer, if I'm not wrong.
 
  • #4
mathics said:
Yes. If the triangle has integer sides and integer heights (the heights also must multiples not primes), then I believe the area should also be integer, if I'm not wrong.
Can you state precisely what you are tryuing to prove?
 
  • #5
PeroK said:
Can you state precisely what you are tryuing to prove?
I'm trying to prove that if the lengths of the sides of a triangle and the heights drawn to all sides are integers, then these integers are all multiples.
 
  • #6
mathics said:
I'm trying to prove that if the lengths of the sides of a triangle and the heights drawn to all sides are integers, then these integers are all multiples.
What does "heights drawn to all sides" mean? Perhaps you could post an image?
 
  • #7
PeroK said:
What does "heights drawn to all sides" mean? Perhaps you could post an image?
Like so.
1709900199205.png
 
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  • #8
I mean, there are some obvious counter examples if you know your Pythagorean triples …
 
  • #9
It seems you could try using the formula that uses the Cross-Product, using coordinates centered at the origin. But, like I think Oro was suggesting, funding a Pythagorean triple with sides a,b both odd, though I think there aren't such pairs/triplets.
 
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  • #10
mathics said:
Like so.
View attachment 341452
So, you want to prove that ##a, b, c, g, i, k## have some relationship?
 
  • #11
mathics said:
Yes. If the triangle has integer sides and integer heights (the heights also must multiples not primes), then I believe the area should also be integer, if I'm not wrong.
The converse is not true however. There are Heronian triangles whose heights are not (all) integers. Examples include some well known Pythagorean triples.
 
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  • #12
Orodruin said:
The converse is not true however. There are Heronian triangles whose heights are not (all) integers. Examples include some well known Pythagorean triples.
Thank you for correcting me.
 
  • #13
PeroK said:
So, you want to prove that ##a, b, c, g, i, k## have some relationship?
Yes, I want to prove if sides a, b, c are integers and heigths k, i, j are also integers, then theese integers are all multiples.

I believe I have to use triangle area formula and Pythagorean theorem, then compare the sides and heights of the triangle.

I would start by selecting one side and proof by contradiction. (E.g I choose side "a" wlog, and say, it has a prime length, and then i would get a contradiction, so the first statement must be true.)

However, I'm not sure how to format it nicely.
 
  • #14
mathics said:
Yes, I want to prove if sides a, b, c are integers and heigths k, i, j are also integers, then theese integers are all multiples.
You mean they are all multiples of some common integer?
 
  • #15
PeroK said:
You mean they are all multiples of some common integer?
Well, all sides can be different, it is just that they have to they must have an integer value and they must be multiple numbers. (For example like 6 is integer number and is also a multiple number.)
 
  • #16
mathics said:
Well, all sides can be different, it is just that they have to they must have an integer value and they must be multiple numbers. (For example like 6 is integer number and is also a multiple number.)
If you mean not prime, then composite is the correct term.
 
  • #17
PeroK said:
If you mean not prime, then composite is the correct term.
Thank you.
 

FAQ: Integer triangle with integer heights

What is an integer triangle with integer heights?

An integer triangle with integer heights is a triangle where all three side lengths are integers, and the perpendicular heights from each vertex to the opposite side are also integers.

How do you determine the height of an integer triangle?

The height of an integer triangle can be determined using the formula for the area of a triangle. For a triangle with sides a, b, and c, and the height corresponding to side a being h_a, the area can be given by both Heron's formula and the basic area formula (1/2) * base * height. By equating these two, we can solve for the height h_a in terms of a, b, and c.

Are there any specific conditions for the existence of integer triangles with integer heights?

Yes, for a triangle to have integer heights, the sides must satisfy certain number-theoretic conditions. Specifically, the area of the triangle, when expressed using Heron's formula, must be an integer, and this area must be divisible by each of the sides to ensure the heights are integers.

Can you provide an example of an integer triangle with integer heights?

An example of an integer triangle with integer heights is the triangle with sides (13, 14, 15). In this triangle, the heights corresponding to these sides are also integers.

Why are integer triangles with integer heights of interest in mathematics?

Integer triangles with integer heights are of interest in number theory and geometry because they represent a special intersection of discrete mathematics and geometric properties. They are useful in various mathematical problems and proofs, and they also have applications in areas such as cryptography and computer science.

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