Find the volume of the pyramid given dihedral angles and the side lengths of the base triangle 
The trianglural pyramid's base has sides 5, 6 and 7. All of the sides of the pyramid make a 60 degree angle with the base. Find the volume of the pyramid.

This was suggested to me by my student, but I am not even sure where to begin. Any hints would be appreciated.

The perpendicular from the top $D$ to the base plane $ABC$, is a point $P$. In the right angle triangle $\triangle DPA$, the angle between $PA$ and $DA$ is $60^\circ$, so $$PA=\frac{DP}{\tan 60^\circ}$$
Similarly, $PB=PC=PA$. Therefore $P$ is the circumcenter. Find it, calculate the height $PD$ from the above formula, and then you get the volume.
