Let A be a corner of a cube. Let B and C be the midpoints of two edges in the positions shown on the figure below:
The intersection of the cube and the plane containing $A, B$, and $C$ is some polygon, $\mathcal{P}$.
(a) How many sides does \mathcal{P} have? Justify your answer.
(b) Find the ratio of the area of \mathcal{P} to the area of \triangle A B C and prove that your answer is correct.
