19. 1 Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ of $\triangle ABC$, respectively, and $AF: FB = BD: DC = CE: EA = 1: m$. Let $AD \cap BE = N$, $BE \cap CF = M$, $CF \cap AD = P$. Find the ratio of the area of $\triangle MNP$ to the area of $\triangle ABC$.
19.1 The area ratio is an affine invariant. Suppose $\triangle ABC$ is an equilateral triangle. With the centroid of the equilateral triangle $ACB$ as the center, rotate $120^{\circ}$, then $A, B, C$ will transform to $B, C, A$, and $D, E, F$ will transform to $E, F, D$. The intersection points of the lines $M, N, P$ will transform to $N, P, M$, thus $\triangle MNP$ is also an equilateral triangle. To find the area ratio of the two equilateral triangles $MNP$ and $ABC$, it suffices to find the ratio of their side lengths. Let $BC=1+m$, and the final calculation yields the area ratio as $(m-1)^{2}:\left(m^{2}+m+1\right)$. This is also the area ratio for the general triangle case.