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        riddles >> medium >> Heron triangle
        
(Message started by: Christine on Apr 24^th, 2014, 8:02pm)

Title: Heron triangle
Post by Christine on Apr 24^th, 2014, 8:02pm

a, b, c are the sides of Heronian triangle.
a and b are odd square numbers.
then GCD(a, b, c) = 1

is it always true? can you provide a counterexample?

Title: Re: Heron triangle
Post by towr on Apr 24^th, 2014, 10:05pm

25,25,30

Title: Re: Heron triangle
Post by Christine on Apr 25^th, 2014, 9:23am

a,b,c are the sides of a Heronian triangle

two sides are square numbers:
a, b are odd squares: (25, 25, 30)
a is even, b is odd : (16, 25, 39)

can we find a Heronian triangle whose two sides are even square numbers?

Is it possible to find a primitive Heronian triangle in which each side is a square number?

Title: Re: Heron triangle
Post by towr on Apr 25^th, 2014, 2:20pm

on 04/25/14 at 09:23:55, Christine wrote:

can we find a Heronian triangle whose two sides are even square numbers?

Sure, just multiply by 4. It preserves squareness and adds evenness. So e.g. 100,100,120

Title: Re: Heron triangle
Post by Christine on Apr 25^th, 2014, 5:02pm

can we find a Heronian triangle whose two sides are even square numbers?

I'm sorry, I forgot to mention a primitive Heronian triangle.

Title: Re: Heron triangle
Post by dudiobugtron on May 22^nd, 2014, 2:02pm

on 04/25/14 at 17:02:23, Christine wrote:

I'm sorry, I forgot to mention a primitive Heronian triangle.

You can't find a primitive Heronian triangle with both smaller sides even, because the hypotenuse will also be even.