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The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger , who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality relating the side lengths ...
This theorem is known as power of a point. A convex quadrilateral is orthodiagonal (has perpendicular diagonals) if and only if the midpoints of the sides and the feet of the four altitudes are eight concyclic points, on what is called the eight-point circle.
Krein–Milman theorem: A convex polygon is the convex hull of its vertices. Thus it is fully defined by the set of its vertices, and one only needs the corners of the polygon to recover the entire polygon shape. Hyperplane separation theorem: Any two convex polygons with no points in common have a separator line. If the polygons are closed and ...
The Archimedean solids are the class of thirteen polyhedrons whose faces are all regular polygons and whose vertices are symmetric to each other; [a] their dual polyhedrons are Catalan solids. [21] The class of regular polygonal faces polyhedron are the deltahedron (whose faces are all equilateral triangles and Johnson solids (whose faces are ...
The Jordan curve theorem can be proved from the Brouwer fixed point theorem (in 2 dimensions), [4] and the Brouwer fixed point theorem can be proved from the Hex theorem: "every game of Hex has at least one winner", from which we obtain a logical implication: Hex theorem implies Brouwer fixed point theorem, which implies Jordan curve theorem.
Centroid of a triangle. In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure.
The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram. [3] The result generalizes to any 2n-gon with opposite sides parallel. Since the sum of distances between any pair of opposite parallel sides is ...
A spiral similarity taking triangle ABC to triangle A'B'C'. Spiral similarity is a plane transformation in mathematics composed of a rotation and a dilation. [1] It is used widely in Euclidean geometry to facilitate the proofs of many theorems and other results in geometry, especially in mathematical competitions and olympiads.