KöMaL 2016 B4774

The parabolas $p_1$ $\big(y=-x^2+b_1 x+c_1\big)$ and $p_2$ $\big(y=-x^2+b_2 x+c_2\big)$ are tangent to the parabola $p_3$ $\big(y=x^2+b_3x+c_3\big)$. Prove that the line connecting the points of tangency is parallel to the common tangent of $p_1$ and $p_2$.