Editing Symmedian (section)

{{Short description|Reflection of a triangle vertex's median over its angle bisector}}
[[Image:Lemoine punkt.svg|thumb|upright=1.25|
{{legend-line|solid grey|[[Median (geometry)|Median]]s (concur at the [[centroid]] {{mvar|G}})}}
{{legend-line|dashed grey|Angle bisectors (concur at the [[incenter]] {{mvar|I}})}}
{{legend-line|solid red|Symmedians (concur at the [[symmedian point]] {{mvar|L}})}}]]

In [[geometry]], '''symmedians''' are three particular [[straight line|lines]] associated with every [[triangle]]. They are constructed by taking a [[Median (geometry)|median]] of the triangle (a line connecting a [[Vertex (geometry)|vertex]] with the [[midpoint]] of the opposite side), and [[reflection (mathematics)|reflecting]] the line over the corresponding [[angle bisector]] (the line through the same vertex that divides the angle there in half). The angle formed by the '''symmedian''' and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.

The three symmedians meet at a [[triangle center]] called the [[Lemoine point]]. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".<ref name="h">{{citation|first=Ross|last=Honsberger|authorlink=Ross Honsberger|contribution=Chapter 7: The Symmedian Point|title=Episodes in Nineteenth and Twentieth Century Euclidean Geometry|publisher=[[Mathematical Association of America]]|location=Washington, D.C.|year=1995}}.</ref>