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In Euclidean geometry, the Simson line theorem states that the three closest points from a point on the circumcircle on to sides (or the extensions of) , and are collinear.
The converse is also true: if the closest points from the three sides of a triangle to a point is collinear, then lies on the circumcircle of the triangle.
Let be a point on the circumcircle of Points and are the feet of the perpendicular bisectors from to sides (or the extensions of) , and respectively. Then and are collinear.
The trilinear equation of the Simson line for a point lying on the circumcircle satisfies .
The Simson line bisects the line , where is the orthocenter.
Moreover, the midpoint of lies on the nine-point circle.
The Simson lines of two opposite point on the circumcircle of a triangle are perpendicular and meet on the nine-point circle.