Concurrence

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

Several lines (or curves) are said to concur at a point if they all contain that point.

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.

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.

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