Ceva's theorem

Ceva's theorem states that three cevians of a triangle are concurrent if and only if the product of their directed side-division ratios is 1.

Cevians: A cevian is a line segment joining a triangle vertex to a point on the opposite side or its extension. The theorem tests when three such vertex-to-side lines meet at one point.
Triangle: A triangle is a polygon with three sides, three vertices, and three interior angles. Ceva's theorem is formulated from its vertices and opposite sides.
Centroid: The centroid of a triangle is the common intersection of its three medians. Each median bisects its opposite side, so the three equal division ratios satisfy Ceva's condition.
Giovanni Ceva: Giovanni Ceva was an Italian mathematician whose 1678 work established the theorem now bearing his name. His published proof made the concurrence criterion part of European geometry.
Directed segment: A directed segment is a segment assigned an orientation, so its length is positive or negative according to direction. Signed side lengths make the product criterion valid when division points lie on side extensions.
Ratio: A ratio compares two quantities by division. Each factor in Ceva's condition compares the two directed portions of a triangle side.
Incenter: The incenter of a triangle is the common intersection of its three internal angle bisectors. The angle-bisector theorem gives side ratios whose product verifies concurrency.
Brianchon's theorem: Brianchon's theorem states that the three diagonals joining opposite vertices of a hexagon circumscribed about a conic are concurrent. It is a later projective dual counterpart to Pascal's theorem and belongs to the broader lineage of incidence results.
Menelaus's theorem: Menelaus's theorem characterizes when three points on the sides or extensions of a triangle are collinear through a product of directed ratios. It gives a related ratio test for collinearity rather than concurrence.
Concurrency (geometry): Concurrency is the property of two or more geometric lines or curves meeting at one point. The theorem's conclusion is precisely the concurrency of three cevians.

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