Cubic Tangents at Root Midpoints

In our geogebra inset session I was challenged to investigate cubic graphs. Specifically, for any cubic graph that has three distinct real roots....

(1) sketch the graph

(2) draw a tangent to the curve at all the midpoints of the roots

(3) what do you notice

(4) can you prove it?

Spoiler alert... Don't look below!

Proving it algebraically was a challenge initially, I'd recommend looking initially at the case where the midpoint of two of the roots is 0, then extend it. Have fun!