APMO · 6. Triangle geometry: lengths, angles and areas
Areas with common heights, parallel lines and similar triangles, triangle centres with vectors, and trigonometry with careful angle chasing: the core tools for APMO triangle problems.
Turn area and ratio information on cevians into equations and solve them.
Prove similarities with parallel lines, the centroid and midlines, matching vertices correctly.
Use vectors with origin at the circumcentre for $O$, $G$, $H$ and signed distances.
Compute a moving configuration with one angle parameter.
Chase angles with the incentre and orthocentre and justify every position.