Triangle

Mathematical statement

Given three distinct noncollinear points A, B, and C in the plane, the union of segments AB, BC, and CA determines the nondegenerate triangle ABC.

Representative formula (exact TeX)

ABC

Assumptions

• The setting is planar Euclidean geometry, and A, B, and C are distinct noncollinear points.

Symbol scope

• A — vertex A
• B — vertex B
• C — vertex C

Intuition, definition, and misconceptions

Triangle ABC has vertices A, B, and C and sides AB, BC, and CA. Reordering the vertices may change the label, but no unstated property should be inferred from the appearance of a diagram.

If any two points coincide or all three points are collinear, they do not form the nondegenerate triangle of this entry. Merely stating that there are three points is insufficient.

This object first establishes vertices, sides, and opposite-side relations. Classification by sides or angles, the angle-sum theorem, area, congruence, and similarity require additional data and theorems.

Common mistakes include assuming any three points form a nondegenerate triangle, confusing vertex names with side names, or guessing side lengths and angles from drawing proportions.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Points, lines, and planes — A triangle is built from planar points, a noncollinearity condition, and connecting segments.

Verification boundary

Explicit points, sides, and noncollinearity data support role recomputation, but do not formally prove triangle existence, angle-sum, or classification theorems.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Choose a EuclideanGeometry model and bind three distinct noncollinear points, segments, and the nondegenerate-triangle statement.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.