Triangle $\triangle ABC$ has side lengths $AB = 80$, $BC = 45$, and $AC = 75$. The bisector of $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P.$ What is $BP$?
$\textbf{(A)}~18\qquad\textbf{(B)}~19\qquad\textbf{(C)}~20\qquad\textbf{(D)}~21\qquad\textbf{(E)}~22$
Difficulty: 72/100 — Can be approached utilizing analytic geometry or other similar concepts.
Core Concepts: cevians, length chasing, angle bisector
Challenges: Finding the intersection of two different cevian types may be new to a lot of competitors.
Techniques: Employ a coordinate system when dealing with unusual cevians or apply another concept such as mass points.
Error-prone Steps: N/A
Ideal Time: Experienced: ≤ 4-5 min
Intermediate: ≤ 8-10 min
Beginner: would not solve