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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]
Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
Classification of quadrilaterals by their symmetry subgroups. [11] The 8-fold symmetry of the square is labeled as r8, at the top of the image. The "gyrational square" below it corresponds to the subgroup of four orientation-preserving symmetries of a square, using rotations but not reflections. The square is the most symmetrical of the ...
The orange and green quadrilaterals are congruent; the blue is not congruent to them. All three have the same perimeter and area. (The ordering of the sides of the blue quadrilateral is "mixed" which results in two of the interior angles and one of the diagonals not being congruent.)
This is not a cyclic quadrilateral. The equality never holds here, and is unequal in the direction indicated by Ptolemy's inequality. The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem.
Lambert quadrilateral fundamental domain in orbifold *p222 *3222 symmetry with 60-degree angle on one of its corners. *4222 symmetry with 45-degree angle on one of its corners. The limiting Lambert quadrilateral has three right angles, and one 0-degree angle with an ideal vertex at infinity, defining orbifold *∞222 symmetry.