106 — Geometry Problems

This is a tall order, but a great one. 106 Geometry Problems (often referring to the book by Titu Andreescu, Vlad Crisan, and Bogdan Enescu, or the classic "103 Trigonometry Problems" / "106 Geometry Problems" from the AwesomeMath series) is an advanced contest problem book for high school students targeting Olympiads (IMO, USAMO, etc.). Below is a deep, structured guide to mastering these problems—covering prerequisites, strategies for each problem type, key theorems, and a study roadmap.

1. Is This the Right Book for You? Prerequisites:

Solid Euclidean geometry up to AIME level. Familiarity with: similar triangles, cyclic quadrilaterals, power of a point, angle chasing. Basic trigonometry (Law of Sines/Cosines, trig form of Ceva). Vectors/complex numbers (helpful for later problems).

Difficulty: The problems range from medium (AIME #10–15) to very hard (IMO #3/6). Do not start here if you are new to geometry. Begin with something like "Geometry Revisited" (Coxeter) or "Lemmas in Olympiad Geometry" . 106 geometry problems

2. Core Theorems & Tools You Must Internalize Before diving into 106 problems, ensure you can state and prove (or at least apply) these: Angle Chasing & Cyclic Quads

Inscribed angle theorem, Thales' theorem. Cyclic quadrilateral: opposite angles sum to (180^\circ), equal angles from chord. Miquel points, spiral similarity.

Triangle Centers

Centroid, orthocenter, circumcenter, incenter, excenters – their properties. Euler line, nine-point circle. Gergonne point, Nagel point, Brocard points (less common but appear).

Power of a Point & Radical Axis

(PA \cdot PB = PT^2) for tangent. Radical axis of two circles = line of equal power. Radical center of 3 circles. This is a tall order, but a great one

Ceva & Menelaus (including trigonometric form)

Ceva: (\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1). Menelaus for transversals. Trig Ceva: (\frac{\sin\angle BAD}{\sin\angle DAC} \cdots = 1).