Difference between revisions of "Concurrence"
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In analytical geometry, one can find the points of concurrency of any two lines by solving the system of equations of the lines. | 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 [[cevian]]s of a triangle to be concurrent. In particular, the three [[altitude]]s, [[angle bisector]]s, [[median]]s, [[symmedian]]s, and perpendicular | + | [[Ceva's Theorem]] gives a criteria for three [[cevian]]s of a triangle to be concurrent. In particular, the three [[altitude]]s, [[angle bisector]]s, [[median]]s, [[symmedian]]s, and [[perpendicular bisector]]s (which are not cevians) of any triangle are concurrent, at the [[orthocenter]], [[incenter]], [[centroid]], [[Lemoine point]], and [[circumcenter]], respectively. |
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Revision as of 18:49, 23 November 2007
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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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