Book 3. Advanced Olympiad Geometry

An advanced inversion module: choosing the center and radius, images of lines and circles, preservation of angles and tangencies, orthogonal circles, chains of circles, and strong final-level configurations.

An introductory advanced module on projective geometry: central projections, cross-ratio, points at infinity, Desargues, Pascal, degenerate tangent cases, Brianchon as a preview, and the three-fixed-points method.

An advanced module on poles and polars with respect to a circle: chord of contact, coordinate formula, La Hire's theorem, self-polar triangles, hidden polars in tangent problems, and Pascal-Brianchon duality.

An advanced module on the pedal triangle and the Wallace-Simson line: projections onto sides, the circumcircle criterion, direction of the Simson line, and links with the orthocenter and nine-point circle.

An advanced geometry-gems module on special triangle points: the Euler line, nine-point circle, Napoleon theorem, Fermat point, symmedians, the Lemoine point, and a preview of Brocard points.

An advanced module on using sines and cosines in geometry: sine and cosine rules, trig Ceva, trig Menelaus, isogonals, special angles, and ratios through sines.