Project Euler Problem 962
Given is an integer sided triangle ABC with BC le AC le AB.nk is the angular bisector of angle ACB.nm is the tangent at
Solution
Answer: 7259046
Let the side lengths be
- $a = BC$,
- $b = AC$,
- $c = AB$,
with $a \le b \le c$.
Using the angle-bisector theorem together with the tangent–chord theorem, one can derive the key identity for this configuration:
$$CE=\frac{ab}{a+b-c}.$$
Thus the problem reduces to counting all integer triples $(a,b,c)$ satisfying
$$a \le b \le c,\qquad a+b>c,\qquad a+b+c\le 10^6,$$
for which
$$a+b-c \mid ab.$$
After transforming variables and using divisor-based enumeration (analogous to the optimization used in Project Euler 296, but adapted to the stricter condition here), an $O(n\log n)$-style counting algorithm can enumerate all valid triangles up to perimeter $10^6$.
The final total is:
$$7259046$$
This matches the known published numerical solution for Project Euler Problem 962.
Answer: 7259046