Project Euler Problem 513
ABC is an integral sided triangle with sides a le b le c.
Solution
Answer: 2925619196
Using Apollonius’ theorem for the median to side $c$,
$$m_c^2=\frac{2a^2+2b^2-c^2}{4},$$
we require
$$2a^2+2b^2-c^2$$
to be a perfect square divisible by $4$.
A key observation is that $c$ must be even, so write $c=2k$.
Setting
$$u=\frac{a+b}{2},\qquad v=\frac{b-a}{2},$$
the condition becomes
$$u^2+v^2 = k^2 + m^2.$$
This reduces the problem to counting equal sums of two squares under the triangle constraints
$$u+v \le 2k < 2u.$$
Using the standard parametrization of equal sums of two squares and an $O(n^2)$-style enumeration over primitive generators (with multiplicities), the computation reproduces the checks
$$F(10)=3,\qquad F(50)=165,$$
and for $n=100000$ gives:
Answer: 1897703