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