# Thread: One for you, WonderBoy

1. ## One for you, WonderBoy

Right triangle: sides 135-352-377
Isosceles triangle: sides 132-366-366
Both have perimeter = 864 and area = 23760

Find another case where a right triangle and an isosceles triangle have the
same area and perimeter; these rules apply, of course:
- all sides are integers
- new case is primitive (not a multiple of above)

2. ## Working on it Wilmer

Without a computer, just a calculator. Give me a couple of days.

3. Originally Posted by wonderboy1953
Without a computer, just a calculator. Give me a couple of days.
Use computer if you wish.

4. ## This was a toughie

Tried applying Goldbach's Conjecture. No success.

Next I flipped Riemann's Conjecture. No luck here.

I next tried a backward FLT with Selmer Groups and a little Galois's Group mixed in. Nothing doing here either.

I guess we'll have to settle for Right Triangle: 270, 704, 754 and
Isosceles Triangle: 264, 732, 732 with...

perimeter of 1728 and area equaling 95,040.

5. Originally Posted by wonderboy1953
I guess we'll have to settle for Right Triangle: 270, 704, 754 and
Isosceles Triangle: 264, 732, 732 with...

6. ## Back to the drawing board

I'll keep trying.

7. ## If this really requires a computer

Then I will decline this challenge until I think of a shortcut or can run a program on a computer to solve this problem.

8. ## Fyi

I don't believe there is another primitive case.

I discovered this in 2003, and sent it to MathWorld; they show it here:
Heronian Triangle -- from Wolfram MathWorld

9. Very nice. So, can you prove that no other primitive case exists?

In my naivete I was trying to take a "constrained optimization" approach to this problem. It of course didn't work, but I still am not totally clear as to why it didn't work.

10. Originally Posted by Wilmer
I don't believe there is another primitive case.

I discovered this in 2003, and sent it to MathWorld; they show it here:
Heronian Triangle -- from Wolfram MathWorld
Did you run the lengths up to 400,000 on a computer (and how long did that take)?

11. ## Wilmer

Did you know that you can only post problems you already know the answer to in this section? "I don't believe there is another primitive case."

12. YES, I know!
However, this one HAS an answer: "it is not known..."
A bit like the Euler Brick:
Euler Brick -- from Wolfram MathWorld

Perhaps Mr Fantastic can issue a statement as to
"it is not known" being a valid solution HERE.

13. Originally Posted by Danny
Did you run the lengths up to 400,000 on a computer (and how long did that take)?
Yes I did. How long? Forgot!
I just did a run getting ALL right triangles where short leg < 100000:
there are 1,521,629 triangles; took 12 minutes 7 seconds (included
was the checking of each to see if it met the criteria).

I hit on this quite accidentally; like, tried it "just to kill time"!
I was surprised that MathWorld "accepted" it.

My computer "search" is simple enough:
get right triangle sides a,b,c (c being hypotenuse, of course)

I use this triangle as half the isosceles triangle.

p = 2(a + c) ; k = 2ab

I then check if there is a right triangle d,e,f (d<e, f the hypotenuse)
that meets the "same perimeter-area" and integer condition:
d,e = [p^2 + 2k +- SQRT(p^4 - 12p^2 k + 4k^2)] / (4p)
(d being the -, e being the +)

So ONLY 2 variables are looped: a and b.