A criterion for three cevians of a triangle (segments joining each vertex to a point on the opposite side) to be concurrent: for points D on BC, E on CA, F on AB, the cevians AD, BE, CF are concurrent if and only if (BD/DC)·(CE/EA)·(AF/FB)=1, interpreting ratios with directed segments when needed.

Principle

Concurrency of lines from vertices to opposite sides is equivalent to a multiplicative balance of directed segment ratios along the three sides; local ratios encode a global incidence condition.

Demonstration

In triangle ABC pick D on BC, E on CA, F on AB. Compute the directed ratios BD/DC, CE/EA, AF/FB; if their product equals 1, draw AD, BE, CF and verify they meet at a common point (or conversely deduce the product equals 1 from known concurrency). For instance, construct a triangle and select D, E so that AD and BE meet at P, then intersect CP with AB at F and check the product equals 1.

Misapplication

Applying Ceva's formula to points that are not on the sides or to cevians that are not drawn from the vertices (for example using arbitrary transversals), or ignoring sign conventions when points lie on extensions of sides, produces incorrect conclusions about concurrency.

Consequence

Provides an algebraic test for concurrency and a constructive tool: given two cevians intersecting at P, one can compute where the third must hit the opposite side. It yields coordinate and mass‑point techniques for ratio computations and triangle center constructions.

Reversal

The dual statement is Menelaus' Theorem: instead of concurrency of cevians one studies collinearity of three intersection points of a transversal, with a signed product equal to −1; Ceva and Menelaus invert incidence roles between points and lines.

Boundary

Holds in Euclidean plane geometry and in projective contexts with directed ratios; requires nonzero segment lengths and distinct intersection points. It does not apply directly when vertices or intersection points coincide in degenerate ways or in geometries without a notion of signed segment ratio.

Semantic Tension

Tension appears with Menelaus' Theorem and with mass‑point heuristics: Ceva's multiplicative criterion is algebraic and local, while mass‑point methods provide an additive intuition; both describe concurrency but from different computational viewpoints.

Synthesis

Ceva's Theorem characterizes concurrency of three cevians in a triangle by a multiplicative equality of (directed) side ratios, serving as a precise algebraic test and a constructive bridge between local segment data and a global intersection.