1984 AIME Problem 3

Each of the three smaller triangles is similar to $\triangle ABC$

Convert each area ratio into a linear ratio and add the three linear ratios

Let the area of $\triangle ABC$ be $K.$ The three small triangles are similar to $\triangle ABC,$ so their corresponding linear ratios are $$\frac{2}{\sqrt K},\qquad \frac{3}{\sqrt K},\qquad \frac{7}{\sqrt K}.$$ Each such scale factor is also the perpendicular distance from $P$ to one side divided by the altitude to that side. These are the three barycentric area ratios $\frac{[PBC]}{K},$ $\frac{[PCA]}{K},$ and $\frac{[PAB]}{K},$ which add to $1.$ Hence $$\frac{2+3+7}{\sqrt K}=1.$$ Therefore $\sqrt K=12$ and $K=144.$