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An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.
It states that in a cyclic quadrilateral , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference ...
A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d {\displaystyle d} approaches zero, a cyclic quadrilateral converges into a triangle A ′ B ′ C ′ , {\displaystyle \triangle A'B'C',} and the formulas above simplify to the analogous triangle formulas.
An equivalent condition is that the bimedians of the quadrilateral (the diagonals of the Varignon parallelogram) are perpendicular. [ 3 ] A convex quadrilateral with diagonal lengths p {\displaystyle p} and q {\displaystyle q} and bimedian lengths m {\displaystyle m} and n {\displaystyle n} is equidiagonal if and only if [ 4 ] : Prop.1
The theorem can be applied to a complex (self-intersecting) quadrilateral. In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side
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.
Thus, the first two triangles are in the same equivalence class, while the third and fourth triangles are each in their own equivalence class. In mathematics , when the elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation ), then one may naturally split the set S {\displaystyle S} into ...
A complete quadrangle (at left) and a complete quadrilateral (at right).. In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points.