Theorem

Ceva's Theorem

Geometry

Gives a precise ratio condition, on the segments cevians cut from a triangle's sides, for three cevians drawn from the triangle's vertices to be concurrent. Named for Giovanni Ceva, it is a classical tool of triangle geometry.

Facts

Statement

For a triangle with cevians drawn from each vertex to a point on the opposite side, the three cevians are concurrent if and only if the product of the three ratios into which they divide the opposite sides equals one. 1

Proof Year

1678 1

Classification

Statement Form

Identity or Equation 1

Statement Form

Characterization Theorem 1

Connections

In Branch

Geometry

Sources

1. Ceva's Theorem (Wikipedia)

Wikimedia Foundation

Lead section, statement of the concurrency construction
Given a triangle △ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of △ABC), to meet opposite sides at D, E, F respectively.
Lead section, sentence on Ceva's 1678 publication
The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis.

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Ceva's Theorem

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