Simson line

The Simson line is the line containing the feet of the perpendiculars from a point on a triangle’s circumcircle to its three sides or their extensions. The three feet are collinear.

Perpendicular projection: The mapping of a point to its nearest point on a line or plane, along a perpendicular. Each of the three defining points is the perpendicular projection of the circle point onto a side line.
Triangle: A polygon with three sides and three vertices. Its vertices define the circumcircle and its three side lines.
Steiner deltoid: A three-cusped hypocycloid traced by a point on a circle rolling inside a circle three times larger. As the point moves around the circumcircle, its Simson line envelope is a Steiner deltoid.
Robert Simson: A Scottish mathematician of the eighteenth century known for work on Euclidean geometry. The line bears his name, though the result is also associated with later mathematicians.
Pedal circle: The circumcircle of the pedal triangle formed by projections of a point onto a triangle’s side lines. For a general point the three feet form a triangle; on the circumcircle they collapse onto the Simson line.
Circumcircle: The unique circle passing through the three vertices of a nondegenerate triangle. The point must lie on this circle for the three projections to align.
Circumcenter: The point equidistant from the vertices of a triangle and the center of its circumcircle. It determines the circle on which the defining point must lie.
Envelope (mathematics): A curve tangent to every member of a one-parameter family of curves or lines. The moving family of Simson lines has a distinctive envelope.
William Wallace: A Scottish mathematician who worked on geometry and made contributions to the theorem now called Simson’s theorem. Wallace is often credited with discovering the collinearity result independently.
Steiner line: A line associated with a point and triangle through the corresponding projections onto the triangle’s side lines. It is a related projection construction, but its defining point is not constrained to the circumcircle.

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