# Difference between revisions of "Concurrence"

 This is an AoPSWiki Word of the Week for Nov 22-28

Several (that is, three or more) lines or curves are said to be concurrent at a point if they all contain that point. The point is said to be the point of concurrence.

## Proving concurrence

In analytical geometry, one can find the point of concurrency of any two lines by solving the system of equations of the lines. To see if it shares the point of concurrency with other lines/curves requires only to test that point.

Ceva's Theorem gives a criteria for three cevians of a triangle to be concurrent. In particular, the three altitudes, angle bisectors, medians, symmedians, and perpendicular bisectors (which are not cevians) of any triangle are concurrent, at the orthocenter, incenter, centroid, Lemoine point, and circumcenter, respectively.

Concurrence of lines can occasionally be proved by showing that a certain point is a center of a particular homothecy.

## Problems

### Introductory

• Are the lines $y=2x+2$, $y=3x+1$, and $y=5x-1$ concurrent? If so, find the the point of concurrency. (Source)

### Intermediate

• In triangle $ABC^{}_{}$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC^{}_{}$, and $AB^{}_{}$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O^{}_{}$, and that $\frac{AO^{}_{}}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92$, find $\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}$.

(Source)