We use barycentric coordinates: $A = (1:0:0), \ M = (0:1:1)$, therefore $P = (6:7:7)$ by the section formula. Now, $I = (a:b:c)$ is the incentre, and $\infty_{AB} = (1:-1:0)$ is the point at infinity of $AB$. We have that $P, I, \infty_{AB}$ are collinear, therefore, $$ \begin{vmatrix} a & b &c\\ 1 & -1 & 0\\ 6 & 7 & 7 \end{vmatrix} = 0.$$Thus, $-7a - 7b + 13c = 0$. Combining with $a+b+c = 100$, we get that $\boxed{c = 35}$.