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 points of concurrency of any two lines by solving the system of equations of the lines.

Related Theorems

• 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.

Problems

Introductory

• Are the lines , , and concurrent? If so, find the the point of concurrency. (Source)

Olympiad

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