Lesson 5 — Ceva’s Theorem

Big idea: Three cevians of a triangle meet at one point exactly when the product of the three side ratios equals 1.

Words to know

Word	Meaning
Cevian	A segment joining a vertex of a triangle to a point on the opposite side.
Concurrent	Three or more lines are concurrent iff they contain the same point.
Ratio	A comparison of two quantities by division.
Opposite side	The side of a triangle that does not contain the chosen vertex.

A cevian goes from a vertex to the opposite side.
Cevians AY, BZ, and CX divide the sides into six segments: AX, XB, BY, YC, CZ, and ZA.
Theorem 73 (Ceva’s Theorem): Three cevians AY, BZ, and CX of △ABC are concurrent iff (AX/XB)(BY/YC)(CZ/ZA) = 1.
The theorem works in both directions: concurrent → product 1, and product 1 → concurrent.
Ceva’s Theorem can prove that the medians are concurrent and that the angle bisectors are concurrent.
The proof uses similar triangles formed by drawing lines parallel to one cevian.
The converse is proved by showing that a second intersection point must be the same point.

(AX/XB) · (BY/YC) · (CZ/ZA) = 1.
Equivalent form: AX · BY · CZ = XB · YC · ZA.
Each ratio compares the two pieces of one side, in the order vertex → point → vertex.

If the homework asks…	Do this…
Test whether three cevians are concurrent	Multiply the three side ratios; concurrent iff the product is 1.
Find a missing ratio	Multiply the two known ratios, then take the reciprocal.
Name the six segments	Start at each vertex and read around the triangle.
Prove medians are concurrent	Each ratio is 1, so the product is 1.
Set up the ratios	Match each cevian to the side it hits, not to another cevian.
Check your product	Cancel before multiplying large numbers.