SMO Senior · 9. Areas and coordinate geometry
Area ratios from common heights and common angles, similar triangles from parallel lines, coordinates for lines, distances and circles, and choosing your own coordinates for hard ratio problems.
Compute unknown areas using common heights, common bases and common angles.
Use parallel lines and midpoints to find congruent and similar triangles and scale areas by $k^2$.
Find intersection points, distances and areas with coordinates.
Read a circle from its equation and find extreme distances and areas with $d\pm r$.
Solve ratio-only problems by choosing convenient coordinates.