The altitudes of a triangle are $15$, $21$, and $35$. What is the largest angle in this triangle?

1. If the altitudes are $h_a,h_b,h_c$, the sides are proportional to $\frac1{h_a}:\frac1{h_b}:\frac1{h_c}$.
2. Here $\frac1{15}:\frac1{21}:\frac1{35}=7:5:3$ (multiplying by $105$).
3. By law of cosines, the angle opposite the side of ratio $7$ has cosine $\frac{5^2+3^2-7^2}{2\cdot5\cdot3}=\frac{-15}{30}=-\frac12$.
4. The largest angle is $\boxed{120^\circ}$.