Euler's inequality
Euler's inequality states that a triangle's circumradius is at least twice its inradius: R ≥ 2r. Equality holds exactly for an equilateral triangle.
Circumradius: The radius of the circle passing through all vertices of a triangle. It is the outer radius R whose lower bound the inequality gives.
Euler's formula for a triangle: The relation R = abc/(4K) between a triangle's circumradius, side lengths, and area. Combined with K = rs, it turns the radius inequality into an algebraic bound.
Degenerate triangle: A limiting triangle whose vertices lie on one line and whose area is zero. The stated inequality concerns nondegenerate triangles, where both radii are defined.
Euler's theorem on the circumcenter and orthocenter: The relation between a triangle's circumcenter, orthocenter, and the midpoint of their joining segment. It is another classical triangle theorem named for Euler, but concerns centers rather than radii.
Inradius: The radius of a triangle's inscribed circle, tangent to all three sides. It is the inner radius r compared with the circumradius.
Heron's formula: A formula giving a triangle's area from its three side lengths and semiperimeter. It can express the area term used to compare the two radii.
Isosceles triangle: A triangle with at least two equal side lengths. Its symmetry makes the relationship between its inradius and circumradius easy to inspect.
Euler line: The line containing a triangle's circumcenter, centroid, and orthocenter. It connects the circumcenter to another central feature of triangle geometry.
Circumcircle: The unique circle passing through the three vertices of a nondegenerate triangle. Its radius supplies one of the two quantities in the inequality.
Semiperimeter: Half the sum of the side lengths of a polygon. The inradius-area identity uses the triangle's semiperimeter as a factor.