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A geometric proof is a step-by-step argument that demonstrates the truth of a statement using definitions, postulates, and previously-proven theorems. The Geometric Proofs Calculator takes a "given" and a "to prove" statement and produces a complete two-column proof — step, reason, step, reason — using whichever postulates and theorems apply. This tutorial explains the structure of a two-column proof, what counts as a valid reason, and the most common proof types you'll encounter in geometry.

The two-column proof format

The traditional format for high-school geometry proofs has two columns:

Every step must have a justification on the right side. Acceptable justifications:

• Given — stated in the problem
• Definition — by the definition of a term (e.g. "definition of midpoint")
• Postulate — a fundamental assumption that doesn't need proof (e.g. SSS Postulate)
• Theorem — a previously-proven statement (e.g. "Vertical Angles Theorem")
• Property — an algebraic property (reflexive, symmetric, transitive, substitution, distributive)
• CPCTC — Corresponding Parts of Congruent Triangles are Congruent (used after proving two triangles congruent)

How the calculator works

Behind the scenes, the calculator uses a large language model trained on thousands of geometry proofs. You provide:

1. The given: what conditions you start from. Example: "AB = CD, BC = DE, ABCD is a quadrilateral".
2. The to prove: the statement you want to demonstrate. Example: "Triangle ABE ≅ Triangle CDE".
3. Optional: a figure photo. The AI Vision system can read both the diagram and any printed labels.

The AI:

1. Parses the given conditions and goal.
2. Identifies which theorem(s) connect them.
3. Builds the chain of statements, citing each justification.
4. Outputs the proof in two-column format (or paragraph form on request).

Standard proof types

Most introductory geometry proofs fall into one of these categories:

1. Triangle congruence proofs

Use one of the 5 congruence postulates (SSS, SAS, ASA, AAS, HL) to prove two triangles congruent. Then CPCTC to extract specific corresponding-part equalities.

Typical structure: identify shared elements (reflexive sides, vertical angles, alternate interior angles), match them up, invoke the postulate, conclude congruence.

2. Triangle similarity proofs

Use one of the 3 similarity postulates (AA, SSS-sim, SAS-sim) to show two triangles similar. Then use proportionality of corresponding sides to derive specific ratios.

3. Parallel-line angle proofs

Establish that two lines are parallel by showing one of the equivalent angle conditions (corresponding equal, alternate interior equal, co-interior supplementary). Or use already-parallel lines to derive equal angles.

4. Quadrilateral classification proofs

Show that a quadrilateral is a parallelogram, rhombus, rectangle, square, kite, or isosceles trapezoid by demonstrating its defining properties.

Example: "Prove ABCD is a parallelogram." Strategy: show both pairs of opposite sides parallel, OR both pairs of opposite sides equal, OR both pairs of opposite angles equal, OR diagonals bisect each other — any ONE of these is sufficient.

5. Circle theorem proofs

Inscribed-angle theorem, tangent-line properties, chord-and-arc relationships, cyclic-quadrilateral angle theorems.

6. Segment and angle proofs

Bisectors, midpoints, perpendiculars, angle additions/subtractions. Often use algebraic properties (substitution, transitive) alongside geometric ones.

What makes a proof "rigorous"

A proof is rigorous when every step is justified by a previously-established statement — no leaps of intuition, no "obviously this is true". Standard high-school geometry grading rubrics expect:

• Each step numbered.
• Each step justified by name (e.g. "Vertical Angles Theorem", not "obvious").
• The progression logical — each step follows from prior ones via the cited theorem/property.
• The final step matches the "to prove" exactly.

Worked example

Given: AB ∥ CD; AB = CD.
To Prove: △ABE ≅ △CDE (where E is the intersection of diagonals AC and BD).

The proof is 5 lines, each justified, leading from the given to the conclusion.

Tips for writing proofs by hand

• Start with given, end with goal. Make sure step 1 cites a "given" and the last step matches "to prove" exactly.
• Identify shared elements early. A shared side or shared angle (reflexive) is often a free step that connects the two parts of your proof.
• Look for parallel lines. They give you many angle equalities "for free" via the parallel-line theorems.
• Don't skip steps. Even algebraically obvious steps like substitution need to be cited. "A = B, B = C, therefore A = C" is three steps, not one.
• Write CPCTC, not "corresponding parts". The standard acronym is universally accepted.

When to use AI vs do it by hand

The AI is fastest for:

• Verifying you've found a correct proof (compare your work to the AI's).
• Generating a proof when you're stuck and need a starting strategy.
• Translating a textbook proof from paragraph form to two-column form (or vice versa).
• Reading a problem off a photo and getting an instant proof.

Do it by hand when:

• It's graded work and the teacher requires it.
• You're studying for an exam (writing proofs by hand cements the patterns).
• The proof is short — for 3-line proofs, AI is overkill.

Limitations

• AI may cite the wrong theorem name. The reasoning is usually correct but the explicit name (e.g. "Vertical Angles Theorem" vs "Linear Pair Theorem") can be inconsistent. Read critically.
• Long multi-stage proofs may be summarized. A proof requiring 12+ steps may be compressed to 6-7. If you need every step, ask for "complete two-column proof, no summarized steps".
• Non-Euclidean geometry not supported. The AI assumes standard Euclidean axioms. Spherical / hyperbolic / projective geometry proofs are out of scope.