SMO Senior · 8. Triangles, similarity and lengths
Compute lengths in triangles with Pythagoras, areas, similar triangles, the angle bisector theorem, midlines and the centroid — and prove simple ratio facts.
Use Pythagoras and the area written twice to find altitudes and projections in right triangles.
Recognise similar triangles from parallel lines (Thales, hourglass) and write correct ratios.
Apply the angle bisector theorem and chain ratios along parallel lines.
Create isosceles triangles with midpoints and midlines to solve angle-relation problems.
Write a complete proof involving the centroid and the $2:1$ ratio on a median.