Bottema's theorem

Bottema's theorem states that three perpendiculars constructed from points on the sides of a triangle are concurrent when the points satisfy a specific relation.

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Triangle: A polygon with three sides and three interior angles. The theorem’s perpendiculars are built from points on a triangle’s sides.
Altitude (triangle): A perpendicular from a triangle vertex to the line containing the opposite side. Altitudes are the best-known perpendiculars associated with a triangle’s sides.
Oene Bottema: A Dutch mathematician known for work in geometry and mechanics. The theorem is named for Bottema, whose geometric writings helped circulate results of this kind.
Miquel's theorem: A theorem asserting concyclicity of four intersection points formed by three lines in a triangle configuration. It is another classical triangle result where local constructions force a global incidence.
Perpendicular line: A line that meets another line at a right angle. Each of the three constructed lines is perpendicular to a side.
Perpendicular bisector: A line perpendicular to a segment through its midpoint. It illustrates how a perpendicular condition can encode a precise geometric constraint.
Triangle geometry: The study of properties and constructions involving triangles and related figures. Bottema’s theorem is one result within this established subject.
Nine-point circle: The circle through a triangle’s side midpoints, altitude feet, and altitude midpoints. It gathers several perpendicular constructions associated with the same triangle.
Ceva's theorem: A theorem stating that three lines through a triangle’s vertices are concurrent exactly when a product of ratios equals one. It provides a standard ratio criterion for concurrency, though Bottema’s construction uses perpendicularity.
Power of a point: A quantity relating a point’s position to the products of secant-segment lengths through a circle. Power-of-a-point identities can convert geometric incidences into length relations.