Ceva theorem

This theorem gives a condition for three lines to be concurent in a triangle.

AM, BN, CP concurrent <=> MB/MC × NC/NA × PA/PB = -1
The ratio are signed values : MB/MC > 0 if M outside BC, < 0 if M between B and C.

An elementary proof results from noting that ratio NC/NA equals ratio of distances from C and A to line BN.
Triangles BQA and BQC, with same base BQ, have then their areas in the same ratio.
That is : NC/NA = (BQC)/(BQA).
Similarily MB/MC = (AQB)/(AQC) and PA/PB = (CQA)/(CQB)
Hence MB/MC × NC/NA × PA/PB = 1 in absolute value, and equals -1 taking the signs of ratios.

Conversly let P' the intersection point of AB and CQ, we get MB/MC × NC/NA × P'A/P'B = -1 from the direct theorem. Hence P'A/P'B = PA/PB then P and P' are the same point.

Application : Gergonne point, Nagel point

Given a triangle, its incircle touching the sides in M, N, P.
AP = AN, BP = BM and CM = CN, hence MB/MC × NC/NA × PA/PB = -1
Lines AM, BN and CP then intersect in one point : the Gergonne point.

Similarily with contact points of excircles.
From the length of common tangents we easily deduce AQ = BP, AR = PC and BR = QC
For instance AU = AV that is AB + BP = AC + CP = AC + (BC - BP) hence BP = (AC + BC - AB)/2
Similarily BA + AQ = BC + CQ = BC + (AC - AQ) hence AQ = (BC + AC - BA)/2, that is  BP = AQ 

Immediately results that AP, BQ and CR intersect in one point : the Nagel point.

Angle variant

Area of BQA = 1/2 BQ.BA.sin(QBA) and similarily for other triangles.
Finally results :

AM, BN, CP concurrent <=>
sin(NBA)/sin(NBC) × sin(PCB)/sin(PCA) × sin(MAC)/sin(MAB) = -1
With the same sign conventions.

Application : symmedians, Lemoine point

Let AM a median and AI the angle bissector of A, AU the symetric of the median from the angle bissector. Lines AM and AU are said "isogonal" : angles BAU = MAC and also BAM = UAC.
AU is then called a "symmedian".
sin(UAB)/sin(UAC) = 1 / ( sin(MAB)/sin(MAC) ) hence
Lines AM... concurrent ⇔ lines AU... also concurrent.

Three lines are concurrent if and only if their isogonal are concurrent

Specifically the symmedians intersect in one point : the Lemoine point.


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