1. Let $A D, B E, C F$ be the three altitudes of an acute triangle $A B C$, with the coordinates of $D, E, F$ being $(4,0),\left(\frac{80}{17}, \frac{20}{17}\right),\left(\frac{5}{2}, \frac{5}{2}\right)$, respectively. Find the coordinates of $A, B, C$.
1. Since $k_{D F}=-\frac{5}{3}=-k_{D E}$, then $A D$ bisects $\angle E D F$. Therefore, $A D$ is perpendicular to the $x$-axis, i.e., the equation of $A D$ is $x=4$.

Given $k_{E F}=-\frac{3}{5}$, let the slope of line $C F$ be $k$. Then, by $C F$ bisecting $\angle D F E$, we get $k= \pm$ 1, so $k_{C F}<-1$. Thus, line $A F$ passes through point $B$, $B(0,0), A(4,4)$.
Since $C$ is on the $x$-axis, let $C(x, 0)$, then by $k_{C F}=-1$ we get $x=5$. Therefore, $c<5$.