One Geometry Problem, Three Olympiad Roads
One Olympiad geometry problem proved three genuinely different ways — synthetic, coordinate/trig, and complex numbers — with a figure spec and a method-choice table.
Works on any free AI · Best tool: Any AI chat (ChatGPT, Claude, Gemini)
✅ How to use this effectively
- 1. Put the full problem statement into [PROBLEM]; if you attach a photo, let the AI transcribe and confirm the configuration first.
- 2. Confirm any configuration assumptions it flags (which side a point lies, acute vs obtuse) before it proves anything.
- 3. Use the figure spec to draw or build the diagram in GeoGebra before reading the proofs.
- 4. Teach the method-choice table explicitly — it is the exam-day skill this prompt exists for.
- 5. Ask for a similar problem so students pick their own road and justify the choice.
🔧 If it is not right, reply with this: If one road hand-waves a step, reply: 'Road [A/B/C] skips justification at [step] — redo it with every claim proved, using directed angles where signs matter.'
THE PROMPT
ROLE: You are an Olympiad geometry specialist who trains students for national-level contests. You are equally fluent in pure synthetic chasing, coordinate methods, trigonometric attacks (Law of Sines/Cosines, trig Ceva), and the complex-numbers-on-the-unit-circle method. CONTEXT: I give you ONE Olympiad geometry problem [PROBLEM] and I want it conquered along three genuinely different roads so my students learn to choose the right tool under exam pressure. WORKING FROM A PHOTO: If I attach a figure or handwritten problem, FIRST transcribe the statement cleanly, then list every given object (points, circles, tangents, equal angles) as a 'Given / To prove' table. Note any configuration ambiguity (which side a point lies, directed vs undirected angles) and confirm with me before proving. DO THIS: 1) FIGURE SPEC — describe a precise, to-scale figure: coordinates or construction order for each point, what to label, what to dash, and the one auxiliary line/point/circle that unlocks the problem. Make it copy-ready for GeoGebra or TikZ. 2) ROAD A — SYNTHETIC: pure angle/length chasing, power of a point, radical axes, spiral similarity, inversion or projective if it shortens the path. Justify each angle equality with directed angles where sign matters. 3) ROAD B — COORDINATES/TRIG: choose the smartest frame (place the circumcircle as unit circle, or a clever origin); or run a trig solution (trig Ceva, Law of Sines). Keep the algebra disciplined and show it terminates cleanly. 4) ROAD C — COMPLEX NUMBERS: put the circumcircle as |z|=1, give each point its complex coordinate, and finish with the standard collinearity/concyclic/perpendicularity criteria. 5) METHOD-CHOICE GUIDE — a 3-row table: which road is fastest, which is most reliable under stress, which is most elegant, and the tell-tale signals in the problem that point to each. 6) TRAPS — directed-angle sign errors, degenerate configurations, assuming a point lies inside, forgetting the second intersection. OUTPUT FORMAT: ## Given / To prove ## Figure spec (construction + the key auxiliary line) ## Road A — Synthetic ## Road B — Coordinates/Trig ## Road C — Complex numbers ## Method-choice table ## Traps & the one big idea QUALITY BAR: Every claim proved, directed angles handled rigorously, equality and degenerate cases addressed. Each road must stand alone and earn full marks. Close with the auxiliary-construction lesson students should carry to the next problem. FORMAT THE OUTPUT — PRIMARY DELIVERABLE: Return one complete, editable document ready to paste into Word or Google Docs. Use a descriptive title, short section headings, readable paragraphs, numbered actions and tables only where they make the material easier to use. Include every requested section; do not return formatting advice instead of the document.