Sample · 26-Triangle Geometry
Chapter 26 Overview
Book 6 · Olympiad Geometry · CHAPTER 26: Triangle Geometry
The triangle is the atom of geometry — the simplest shape, yet the source of endless depth. This chapter builds the core toolkit that wins olympiad geometry problems. You will start with angle chasing, tracking angles through triangles and circles until an unknown falls out, then upgrade to directed angles modulo \( 180^\circ \), the professional’s tool that makes configuration issues vanish and proves concyclicity in one line. You will master similarity and ratio chasing, spot hidden congruences unlocked by auxiliary constructions, and compute with area ratios using shared bases and altitudes. Finally you will wield the three great concurrency-and-collinearity theorems: Ceva (in both ratio and trigonometric form) and Menelaus. These techniques underpin everything from AMC angle puzzles to IMO problem 3s, and echo in physics, computer graphics, robotics, and structural engineering — anywhere shapes and proportions matter. The chapter is deliberately diagram-rich and colorful, with labeled figures for every idea, so you can see each theorem as well as prove it. We march through seven Topics — 26.1 Angle Chasing, 26.2 Directed Angles, 26.3 Similarity, 26.4 Congruence, 26.5 Area Methods, 26.6 Ceva’s Theorem, and 26.7 Menelaus’ Theorem — from AMC warmups to IMO-level configurations, each with worked solutions, real-world connections, multiple figures, and eight fully solved problems. Sharpen your compass!
26.1 Angle Chasing
26.1 Angle Chasing
Angle chasing is the art of finding an unknown angle by repeatedly applying a handful of standard relations until the answer appears. The bedrock facts: the angles of a triangle sum to \( 180^\circ \); an exterior angle equals the sum of the two remote interior angles; base angles of an isosceles triangle are equal; and vertical angles are equal. Add the circle relations and you can chase almost anything.
The circle gives the inscribed angle theorem: an inscribed angle is half the central angle subtending the same arc, so all angles inscribed in the same arc are equal. Two consequences dominate: opposite angles of a cyclic quadrilateral sum to \( 180^\circ \), and the tangent-chord angle equals the inscribed angle in the alternate segment. When the chase stalls, an auxiliary angle — a well-chosen extra line or point that introduces a new isosceles triangle or cyclic quadrilateral — often breaks it open.
📋 Standard angle relations
| Relation | Statement |
|---|---|
| triangle sum | \( \angle A+\angle B+\angle C=180^\circ \) |
| exterior angle | \( \angle_{\text{ext}}=\angle A+\angle B \) |
| inscribed angle | inscribed \( =\tfrac12 \) central (same arc) |
| cyclic quadrilateral | opposite angles sum to \( 180^\circ \) |
| incenter / orthocenter | \( \angle BIC=90^\circ+\tfrac{A}{2},\ \angle BHC=180^\circ-A \) |
🌍 Math in the Real World — sightlines and structures
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