RO-NMO 2000 Problem 103

Let be a tetahedron $ABCD,$ and $E$ be the projection of $D$ on the plane formed by $ABC.$ If $\mathcal{A}_{\mathcal{R}}$ denotes the area of the region $\mathcal{R},$ show that the following affirmations are equivalent:

a) $C=E\vee CE\parallel AB$ b) $M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2$