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Concurrent lines arise in the dual of Pappus's hexagon theorem. For each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side. Then the segments connecting the circumcenters of opposite triangles are concurrent. [8]
The line joining them is then called the Pascal line of the hexagon. Brianchon: If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the Brianchon point of the hexagon.
Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC. In Euclidean geometry, Ceva's theorem is a theorem about triangles.
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal.
Given a set of collinear points, by plane duality we obtain a set of lines all of which meet at a common point. The property that this set of lines has (meeting at a common point) is called concurrency, and the lines are said to be concurrent lines. Thus, concurrency is the plane dual notion to collinearity.
Commandino's theorem, named after Federico Commandino (1509–1575), states that the four medians of a tetrahedron are concurrent at a point S, which divides them in a 3:1 ratio. In a tetrahedron a median is a line segment that connects a vertex with the centroid of the opposite face – that is, the centroid of the opposite triangle.
A lot of U.S. history is too good to be true — and actually is not. Sometimes fact is ignored, or teachers miss the latest, and these tales are examples.
A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides 1, 3, 5, … are equal, and sides 2, 4, 6, … are equal). [11] A cyclic pentagon with rational sides and area is known as a Robbins pentagon. In all known cases, its diagonals also have rational lengths, though ...