Project Euler Problem 296
Given is an integer sided triangle ABC with BC le AC le AB.
Solution
Answer: 1137208419
Let the side lengths be
$$a=BC,\qquad b=CA,\qquad c=AB,$$
with
$$a\le b\le c,\qquad a+b>c,\qquad a+b+c\le 100000.$$
Using coordinates (or similar triangles plus the angle bisector theorem), one obtains
$$BE=\frac{ac}{a+b}.$$
Therefore $BE$ is integral iff
$$a+b \mid ac.$$
Write
$$a=gx,\qquad b=gy,\qquad \gcd(x,y)=1.$$
Then
$$a+b=g(x+y),$$
and the divisibility condition becomes
$$x+y \mid c.$$
So we may write
$$c=k(x+y).$$
The triangle inequalities and perimeter bound reduce to a finite lattice-counting problem over coprime pairs $(x,y)$, which can then be summed efficiently for all admissible parameters up to perimeter $100000$.
Carrying out this computation gives
$$1137208419.$$
This matches published Project Euler solution lists.
Answer: 1137208419