SMO Junior · 9. Areas, area ratios and coordinates
Compare areas through common heights, similar triangles and named unknown pieces, and finish hard cases with coordinates.
Turn base ratios into area ratios for triangles with a common height.
Use similar triangles: length ratio $k$ gives area ratio $k^2$.
Set up area equations for crossing cevians and for trapezia with diagonals.
Compute areas with coordinates, the box method and $\frac12|x_1y_2-x_2y_1|$.