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A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square. [7] A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle. [8] A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. [9]
The Van Aubel points, the mid-points of the quadrilateral diagonals and the mid-points of the Van Aubel segments are concyclic. [3] A few extensions of the theorem, considering similar rectangles, similar rhombi and similar parallelograms constructed on the sides of the given quadrilateral, have been published on The Mathematical Gazette. [5] [6]
A pentagon is a five-sided polygon. A regular pentagon has 5 equal edges and 5 equal angles. In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain.
Four line segments, each perpendicular to one side of a cyclic quadrilateral and passing through the opposite side's midpoint, are concurrent. [23]: p.131, [24] These line segments are called the maltitudes, [25] which is an abbreviation for midpoint altitude. Their common point is called the anticenter.
The orange and green quadrilaterals are congruent; the blue one is not congruent to them. Congruence between the orange and green ones is established in that side BC corresponds to (in this case of congruence, equals in length) JK, CD corresponds to KL, DA corresponds to LI, and AB corresponds to IJ, while angle ∠C corresponds to (equals) angle ∠K, ∠D corresponds to ∠L, ∠A ...
Euler's quadrilateral theorem or Euler's law on quadrilaterals, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem .
Another similar characterization concerns the exradii r a, r b, r c, and r d in the same four triangles (the four excircles are each tangent to one side of the quadrilateral and the extensions of its diagonals). A quadrilateral is tangential if and only if [1]: p.70
A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry.The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals.