(A) Prove that a triangle with integer sides cannot have an area that is prime.
(B) Determine a triangle with rational sides whose area and perimeter are both prime.
Assuming this can be derived from a Heronian triangle. If we had one with area n^2*p and perimeter n*q for primes p and q then we could scale it down by a factor of n.
A hand search of the smallest triangles given on
https://en.wikipedia.org/wiki/Heronian_triangle#Properties_of_side_lengths
gives no results.
(But some are close. There's a pair with A=396=36*11 and p=198=6*33 but of course, 33 is not prime.)
There are some nice parameterizations such as Euler's
gcd(m,n)=gcd(p,q)=1
Area=mnpq(mq+np)(mp-nq)
perimeter=2mp(mq+np)
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Posted by Jer
on 2024-06-03 16:34:55 |
How one might computer search part B
