Simson Line Theorem
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In Euclidean geometry, the Simson line theorem states that the three closest points from a point \(P\) on the circumcircle on \(\triangle ABC\) to sides (or the extensions of) \( AB \), \( AC, \) and \( BC \) are collinear.
The converse is also true: if the \(3\) closest points from the three sides of a triangle to a point \(P\) is collinear, then \(P\) lies on the circumcircle of the triangle.
Let \( P \) be a point on the circumcircle of \( \triangle ABC. \) Points \( D, E,\) and \( F \) are the feet of the perpendicular bisectors from \(P\) to sides (or the extensions of) \( AB \), \( AC, \) and \( BC, \) respectively. Then \( D, E,\) and \( F \) are collinear.
The trilinear equation of the Simson line for a point \(p:q:r\) lying on the circumcircle satisfies \( cpq+bpr+aqr=0\).
The Simson line bisects the line \(HP\), where \(H\) is the orthocenter.
Moreover, the midpoint of \(HP\) 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.