Ceva's theorem
Ceva's Theorem
Given triangle ABC, construct a line QR through B, such that QR is parallel to AC. The line AR intersects side BC at X and CQ intersects side AB at Z. Let P be the intersection of AR and CQ. Assume BP intersects side AC at Y. Thus, the Cevians AX, BY, and CZ are concurrent.
We know that QR is parallel to AC. So, by using Euclid's Proposition 29 to find congruent angles we can show that there are four pairs of similar triangles:
AZC~BZQ,
BXR~CXA,
CYP~QBP,
YAP~BRP.
From the similarities, we can get equal ratios:
,
,
,
From the last two equations, we obtain the equal ratios .
Notice that
Ceva's Theorem converse
The contrapositive of the converse of Ceva's Theorem says:
If AX, BY, and CZ are not concurrent, then
Given the triangle construction described in the proof of Ceva's theorem above, assume that AX, BY, and CZ are not concurrent.
Let AX and CZ intersect at a point P. BY does not intersect AX and CZ at P, but construct a new cevian BW such that AX,CZ,BW are concurrent. Then
Notice , thus