https://www.youtube.com/watch?v=aIXcrJur7tU

We have a right triangle, sides 135-352-377.
Your mission:
find an integer-sided isosceles triangle with the
same perimeter and area as above right triangle.

Re: Your Mission. After a frantic and lengthy search in the attic I was lucky enough to find one of those integer-sided things and have now put it on Ebay. Thanks Denis.

Bernie

Ahhh geeee...hope there was no scary giant spiders
in your attic, Bernie...

The sides of the isosceles triangle are: 366, 366, 132

Perimeter of the isosceles triangle = 864
Area of the isosceles triangle = 23760

x = length of one of the equal sides
y = length of the remaining side (base) = 864-2x

Height of the isosceles triangle = sqrt[(x^2 - (432-x)^2}]
= sqrt[x^2 - (432^2 - 864x + x^2)]
= sqrt(864x-432^2)

Area of the isosceles triangle = (1/2)(864-2x)[sqrt(864x-432^2)] = 23760
This simplifies to

x^3 - 1080x^2 + 373248x - 40964184 = 0

No way am I going to solve this without WolframAlpha which gave me x = 366

Keerect CCb!
Being lazy, I often use WolframALpha,
even for simpler quadratics!

I discovered (kinda accidentally) those
2 triangles in 2003; what's unique with
them is they are the only primitive case
of this scenario.

I did wonder why you mention the right triangle at all, since it's not needed.
Now I know, thanks!

I really should have put up the problem this way:

We have an isosceles triangle, sides 132-366-366.
Your mission:
find an integer-sided right triangle with the same
perimeter and area as the above isosceles triangle.

Makes getting the solution more "fun"; no cubic!!

I had fun doing it even with the cubic, Denis.
Solving the other direction looks like fun too.