A foundational projective geometry theorem: two triangles are perspective from a point (corresponding vertices joined by lines meeting at a point) if and only if they are perspective from a line (the three intersection points of corresponding sides are collinear), assuming the usual incidence axioms of a projective plane.

Principle

Incidence relations in projective space obey a duality: pointwise perspectivity and linewise perspectivity are equivalent under projective projection, reflecting an underlying three‑dimensional embedding or coordinatization by a division ring in Desarguesian planes.

Demonstration

Place two triangles in three‑dimensional space with corresponding vertices on three concurrent lines through a point; project them onto a plane to obtain two planar triangles perspective from a point. Compute intersections of corresponding sides and check they lie on a single line, illustrating the theorem. Conversely, from collinearity of those intersections, reconstruct a perspectivity point in a 3D lift when the plane is Desarguesian.

Misapplication

Assuming Desargues' conclusion in an arbitrary incidence structure; the theorem can fail in non‑Desarguesian projective planes, so reasoning that requires Desargues without verifying projective coordinatizability is unwarranted. Also misapplying in degenerate cases where corresponding sides are parallel or intersection points coincide requires careful handling.

Consequence

When valid, Desargues' Theorem ties planar configurations to a three‑dimensional projective viewpoint and justifies many projective constructions and coordinate arguments; it is a litmus test for whether a projective plane is coordinatizable by a division ring (Desarguesian).

Reversal

Inversion swaps the roles of 'perspective from a point' and 'perspective from a line'; the theorem's logical equivalence becomes a bridge between pointwise and linewise incidence. Negating the theorem leads to non‑Desarguesian geometries where such equivalence fails, revealing alternative incidence structures.

Boundary

Holds in all Desarguesian projective planes (those arising from a three‑dimensional vector space over a division ring) and in Euclidean/projective plane models derived from them. It need not hold in exotic projective planes; degeneracies (coincident vertices, parallel corresponding sides interpreted as meeting at infinity) require projective closure and careful treatment of points at infinity.

Semantic Tension

Tension exists between Desargues and more elementary Euclidean intuitions: many Euclidean proofs silently use three‑dimensional embeddings that are not justified in arbitrary projective planes. Also Desargues relates to Pappus: Pappus is stronger (implies commutativity), so satisfaction of Desargues but not Pappus distinguishes algebraic structure of the coordinate ring.

Synthesis

Desargues' Theorem asserts an equivalence between point perspectivity and line perspectivity for two triangles in Desarguesian projective planes, encoding a deep link between planar incidence and three‑dimensional/projective coordinatization while marking a boundary between classical and exotic projective geometries.