Barrow's inequality

Barrow's inequality states that, for a point inside a triangle, the sum of its distances to the vertices is at least the sum of its perpendicular distances to the sides.

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Triangle: A polygon with three sides and three interior angles. The inequality compares distances defined by a triangle’s vertices and sides.
Isaac Barrow: An English mathematician and theologian who held the Lucasian Chair before Isaac Newton. Barrow's name is attached to this triangle inequality.
Geometric inequality: An inequality relating geometric quantities such as lengths, angles, areas, or volumes. Barrow's result is a distance comparison in plane geometry.
Routh's theorem: A formula for the area of the triangle formed by three cevians in a reference triangle. It is another named result in classical triangle geometry, though it concerns area rather than distance sums.
Perpendicular distance: The shortest distance from a point to a line, measured along a perpendicular. Each side contributes the distance from the interior point to its perpendicular foot.
Classical geometry: The study of geometric figures and their properties using Euclidean methods. The inequality belongs to the tradition of synthetic results about triangles.
Distance geometry: The study of geometric structure through distances between points. The inequality bounds one sum of point-to-point distances using point-to-line distances.
Ceva's theorem: A criterion for three cevians of a triangle to meet at one point. It illustrates the broader family of classical theorems organized around triangle points and lines.
Triangle inequality: A geometric rule stating that one side of a triangle is shorter than the sum of the other two. Proofs use it to bound vertex distances through points on the sides.
Euclid's Elements: An ancient Greek mathematical treatise that systematized geometry and number theory. Its triangle principles provide background for classical geometric inequalities.