Apollonius's theorem

In any triangle, the sum of the squares of two sides equals twice the square of the median to the third side plus half the square of that side.

Triangle median: A segment joining a triangle's vertex to the midpoint of the opposite side. The theorem measures the side lengths against the median drawn to the third side.
Triangle: A polygon with three sides and three interior angles. Apollonius's theorem is an identity among the lengths in any triangle.
Triangle side-length calculation: The use of known triangle measurements to determine an unknown side or segment length. Knowing two sides and the median gives the length of the third side directly.
Apollonius of Perga: An ancient Greek mathematician known for his work on conic sections. The theorem bears his name, although its attribution to him is not firmly established.
Parallelogram law: An identity stating that the sum of the squares of a parallelogram's diagonals equals the sum of the squares of its sides. A parallelogram construction turns the median identity into a direct instance of this law.
Midpoint: The point halfway between the endpoints of a segment. The median ends at the midpoint, so the opposite side is split into equal lengths.
Centroid: The point where a triangle's three medians intersect, located two-thirds of the way along each median from its vertex. The median length relation can be combined with the centroid's position to relate vertex distances to the centroid.
Euclid: An ancient Greek mathematician whose Elements systematized geometry and number theory. The theorem is part of the classical geometry tradition represented by Euclid's Elements.
Vector projection: The component of one vector in the direction of another, represented by a scalar or vector. Splitting the two sides into parallel and perpendicular components makes the median formula transparent.
Pythagorean theorem: A right-triangle identity stating that the square of the hypotenuse equals the sum of the squares of the legs. A perpendicular construction proves the median relation by applying this identity twice.

Linked from 1 page