AI 기하학 증명 생성기

The AI Geometry Proof Generator is a specialized tool for producing two-column geometry proofs from a Given and Prove statement. It is the most demanding of the AI tools on this site — generating rigorous proofs requires the AI to identify the right theorems, sequence them logically, and write each justification correctly. This tutorial covers how to write a good prompt, what kind of proofs the AI handles best, the format you get back, and the limitations to watch for.

The two-column proof format

A two-column proof is the standard formal structure taught in high-school geometry. Two columns:

• Statement — what you assert at this step.
• Reason — why this assertion is true (definition, postulate, theorem, or property already established).

Each step is numbered. The proof flows from "Given" at step 1 to "To Prove" at the final step. Every step must have a justification — no jumps, no intuition, no "obviously".

How to write the Given and Prove inputs

The AI does best when both fields are explicit.

Good Given: "Triangle ABC has AB = AC. Point D is the midpoint of BC. Segment AD is drawn." — All facts are named, sides are labeled, the construction is clear.

Bad Given: "Isosceles triangle with a midpoint." — The AI has to guess which point is the midpoint, where AD is, and how the elements relate. The proof will be wobbly.

Good Prove: "AD is perpendicular to BC." — A specific, named geometric claim.

Bad Prove: "Find the relationship." — Vague. The AI may produce a relationship, but not the specific one the textbook wants.

What kinds of proofs work well

The AI Proof Generator handles all of the following reliably:

• Triangle congruence proofs using SSS, SAS, ASA, AAS, HL.
• Triangle similarity proofs using AA, SSS-sim, SAS-sim.
• Parallel-line angle proofs (alternate interior, corresponding, co-interior).
• Quadrilateral classification — proving a quadrilateral is a parallelogram, rhombus, rectangle, kite, or trapezoid.
• Circle theorem proofs — inscribed angles, tangent properties, chord-arc relationships.
• Segment proofs — perpendicular bisectors, angle bisectors, midpoints.
• Algebraic-geometric proofs mixing angle / segment algebra with geometric reasoning.

Worked example — typical input and output

Given: Triangle ABC with AB = AC. AD bisects ∠BAC, with D on BC.

To Prove: AD bisects BC.

Notice the structure: each statement is justified, the chain flows from given to goal, and the final step matches the To Prove exactly.

Paragraph proofs vs two-column

The default output is two-column. To request a paragraph proof instead, add "paragraph proof" anywhere in your input fields. The AI will produce the same logical chain in narrative form — sometimes preferred for higher-level math classes or competition writing.

Flow chart proofs

For complex multi-branch proofs, you can request "flow chart proof" and the AI will return the steps in a visual flowchart format showing the dependency tree of statements.

Common limitations

The AI is not perfect. Watch for these typical mistakes:

• Wrong theorem name. The reasoning may be correct but the cited theorem may be wrong (e.g. "Vertical Angles Theorem" written as "Corresponding Angles Theorem"). Always read the reasons critically.
• Skipped steps. A subtle "obviously" assumption that should have been a separate step.
• Over-explanation. Sometimes the AI inserts unnecessary intermediate steps that aren't wrong but pad the proof. For graded work, ask for "minimum-step proof".
• Non-Euclidean assumptions. The AI assumes standard Euclidean geometry. Spherical / hyperbolic / projective work is out of scope.
• Visualization gaps. If the configuration is ambiguous (e.g. "two intersecting lines" — but which two?), the AI may pick a different interpretation than your figure shows.

When to use this vs the general AI Solver

• AI Proof Generator (this tool): explicitly for formal proofs — given/prove structure, two-column output, theorem citations.
• AI Geometry Problem Solver (AI-01): open-ended word problems, computations, mixed proof/calc problems.
• Geometric Proofs Calculator (AI-06): like this tool but with image upload (for textbook diagrams).

Tips for using the output

1. Read every step. The AI is usually right but occasionally wrong — verifying takes a few minutes per proof.
2. Check that the reasons match standard textbook vocabulary. Some teachers require specific phrasing.
3. If the proof has 12+ steps, ask for it to be condensed. Often the AI takes safer (longer) paths than needed.
4. Compare with the textbook's expected solution if available. The two should match in essence, though wording may differ.

Credits and cost

Each proof generation uses 3 credits. New accounts receive 30 free credits, enough for 10 proofs. See Pricing for higher-volume plans.

Common mistakes when prompting

• Forgetting to label the figure. If your problem has unlabeled points, the AI cannot reference them. Use specific labels in your Given (A, B, C, ..., M, N, P, ...).
• Asking for the answer without the proof. This tool generates PROOFS. For numerical answers without proof, use the general AI Solver.
• Embedding multiple unrelated problems. Each request should focus on one proof. Multi-part proofs work but get sprawly.
• Expecting non-textbook geometry. The AI handles standard high-school + early-college Euclidean geometry. Topology, abstract algebra, advanced number theory are out of scope.