Question 49
Let $ABC$ be a triangle such that $AB = AC = 182$ and $BC = 140$. Let $X_{1}$ lie on $AC$ such that $CX_{1} = 130$. Let the line through $X_{1}$ perpendicular to $BX_{1}$ at $X_{1}$ meet $AB$ at $X_{2}$. Define $X_{2}, X_{3}, \ldots$, as follows: for $n$ odd and $n \geq 1$, let $X_{n+1}$ be the intersection of $AB$ with the perpendicular to $X_{n-1}X_{n}$ through $X_{n}$; for $n$ even and $n \geq 2$, let $X_{n+1}$ be the intersection of $AC$ with the perpendicular to $X_{n-1}X_{n}$ through $X_{n}$. Find $BX_{1} + X_{1}X_{2} + X_{2}X_{3} + \ldots$