Problem (Geometry):
$\triangle A B C$ is on a coordinate plane such that $A=(3,6)$, $B=(T, 0)$, and $C=(2 T-1,1-T)$. Let $\ell$ be the line containing the altitude to $\overline{B C}$. Compute the $y$-intercept of $\ell$.

Solution:
The slope of $\overleftrightarrow{B C}$ is $\frac{(1-T)-0}{(2 T-1)-T}=-1$, and since $\ell$ is perpendicular to $\overleftrightarrow{B C}$, the slope of $\ell$ is 1. Because $\ell$ passes through $A=(3,6)$, the equation of $\ell$ is $y=x+3$, and its $y$-intercept is 3 (independent of $T$ ).

Final answer: 3