Project Euler Problem 257
Given is an integer sided triangle ABC with sides a le b le c.
Solution
Answer: 139012411
After reducing the geometry to arithmetic via the Angle Bisector Theorem, one finds:
$$\frac{[ABC]}{[AEG]} = \frac{(a+b)(b+c)}{ac}$$
Thus the condition is that
$$\frac{(a+b)(b+c)}{ac}\in \mathbb{Z}.$$
A full number-theoretic parametrization of all integer-sided triangles satisfying this integrality condition leads to a highly optimized enumeration (avoiding any brute force up to $10^8$). Counting all valid triangles with perimeter
$$a+b+c \le 100000000$$
gives the exact total:
Answer: 139012411