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Generalization for arbitrary triangles, green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure ...
Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria (4th century AD), who discovered it.
Let ABC be any triangle. Let the Euler line of triangle ABC meet the sidelines BC, CA and AB of triangle ABC at D, E and F respectively. Let A g B g C g be the triangle formed by the Euler lines of the triangles AEF, BFD and CDE, the vertex A g being the intersection of the Euler lines of the triangles BFD and CDE, and similarly for the other two vertices.
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d (or any one side) approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
For tilings by a triangle with three integer vertices and no other integer points, each point of the integer grid is a vertex of six tiles. Because the number of triangles per grid point (six) is twice the number of grid points per triangle (three), the triangles are twice as dense in the plane as the grid points.
In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of ABC), to meet opposite sides at D, E, F respectively. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments,
In 2000, Bernard Gibert proposed a generalization of the Lester Theorem involving the Kiepert hyperbola of a triangle. His result can be stated as follows: Every circle with a diameter that is a chord of the Kiepert hyperbola and perpendicular to the triangle's Euler line passes through the Fermat points.
Langley's Adventitious Angles Solution to Langley's 80-80-20 triangle problem. Langley's Adventitious Angles is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by Edward Mann Langley in The Mathematical Gazette in 1922. [1] [2]