Angle Chasing: Geometry When You Cannot See the Picture
Most olympiad geometry starts with a fact you already know from school. The skill is knowing which fact to apply next — and how to keep the chain short.
Olympiad Desk · Core Learning
What you will take away
- Draw a clean, accurate diagram — most solutions are discovered, not derived
- Cyclic quadrilaterals convert angle equality into a one-line tool
- Isosceles triangles and midpoints pair with the same-angle rule
- Write every conclusion as a labelled statement so the chain stays auditable
Angle chasing is the part of geometry that behaves like algebra: you apply small, legal moves until the target appears. The moves are few, and they are all on your school syllabus.
The four moves
- Angles in a triangle add to 180°, and a straight line adds to 180°
- Equal arcs subtend equal angles: a cyclic quadrilateral gives ∠A + ∠C = 180°
- Isosceles triangles have equal base angles — look for two radii or two tangents
- Parallel lines transfer angles: alternate and corresponding angles are equal
The cyclic quadrilateral is the workhorse
If ∠ABD = ∠ACD then A, B, C, D lie on a circle. Going in the other direction, if they lie on a circle then equal arcs give equal angles for free. This is the single most useful equivalence in olympiad geometry: prove a circle, then harvest equal angles everywhere.
How to write it up
- 1State what you want to prove in angle terms (e.g. ∠X = ∠Y)
- 2List known angle facts about the figure from the statement
- 3For each step, name the rule you are applying and the angle it produces
- 4Finish when the accumulated facts force the target equality
Worked problems
- 1
In triangle ABC, D is the midpoint of BC and AD is perpendicular to BC. Prove that AB = AC.
Show hint
AD is a shared side, and the two right angles are equal — this is a congruent triangles problem in disguise.
Show solution
Triangles ABD and ACD have BD = CD (midpoint), AD = AD (shared) and ∠ADB = ∠ADC = 90°. By SAS they are congruent, so AB = AC.
Answer: ABD ≅ ACD by SAS, hence AB = AC.
- 2
A, B, C, D lie on a circle in that order. Prove that ∠ABD = ∠ACD.
Show hint
Both angles subtend the same arc.
Show solution
Both angles are inscribed angles standing on the chord AD (the arc AD not containing the vertex). Inscribed angles on the same arc are equal.
Answer: They subtend the same arc AD, so the angles are equal.
