Integer triangle with integer heights

[mathics]

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?

 

[PeroK]

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

https://en.wikipedia.org/wiki/Heronian_triangle

 

[mathics]

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.

 

[PeroK]

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?

 

[mathics]

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.

 

[PeroK]

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?

 

[mathics]

What does "heights drawn to all sides" mean? Perhaps you could post an image?

Like so.

 

[Orodruin]

I mean, there are some obvious counter examples if you know your Pythagorean triples …

 

[WWGD]

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.

 

[PeroK]

Like so.
View attachment 341452

So, you want to prove that $a, b, c, g, i, k$ have some relationship?

 

[Orodruin]

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.

 

[mathics]

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.

 

[mathics]

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.

 

[PeroK]

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?

 

[mathics]

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.)

 

[PeroK]

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.

 

[mathics]

If you mean not prime, then composite is the correct term.

Thank you.