Definitions and some Properties of Quadrilaterals

The quadrilaterals below are listed in order from top to bottom, and from left to right in correspondence with the dynamic Hierarchical Quadrilateral Tree where this page is linked from, but the duals of each other are grouped together next to each other to display the side-angle duality. Only a selection of some important properties are given. Since quadrilaterals lower down in the hierarchy inherit ALL the properties from those they are linked to from above, only some additional properties are given lower down. Also note that each of the quadrilaterals below can be mathematically defined in several different, but equivalent ways. So the definitions below, mostly based on symmetry, are not unique.

Quadrilateral - Closed, plane figure with four vertices A, B, C and D, connected by four straight sides AB, BC, CD and DA. Properties: If convex, no reflexive angles, both diagonals interior & interior angle sum is 3600. If concave, one reflexive angle, one diagonal exterior & interior angle sum is 3600. If crossed, two reflexive angles, two diagonals exterior & interior angle sum is 7200.

Circumscribed Quadrilateral - any quadrilateral circumscribed around a circle. Properties:
1) Incircle.
2) Concurrent angle bisectors.
3) Two sums of the pairs of opposite sides are equal.
4) Convex or concave.
5) Area = semi-perimeter times inradius.

Cyclic Quadrilateral - any quadrilateral inscribed in a circle. Properties:
1) Circumcircle.
2) Concurrent perpendicular side bisectors.
3) Two sums of the pairs of opposite angles are equal.
4) Convex or crossed.
5) Convex area = ½sin A*(ad + bc), in sketch.

Perpendicular or Orthodiagonal Quadrilateral - any quadrilateral with perpendicular diagonals. Properties:
1) Intersection with Circum Quadrilateral is a Kite.
2) Intersection with Bisecting Quadrilateral is a Kite.
3) Midpoints of sides form a rectangle.
4) Convex, concave or crossed.
5) Area = ½product of diagonals.

Diagonal or Equidiagonal Quadrilateral - any quadrilateral with equal diagonals. Properties:
1) Intersection with Cyclic Quadrilateral is an Isosceles Trapezoid.
2) Intersection with Trapezoid is an Isosceles Trapezoid.
3) Midpoints of sides form a rhombus.
4) Convex, concave or crossed.

Bisecting or Bisect-diagonal Quadrilateral - any quadrilateral with at least one of its diagonals bisected by the other. Properties:
1) Intersection with Circum Quadrilateral is a Kite.
2) In the sketch, AC which divides BD, also divides the area of ABCD in half.
3) Midpoint N of AC divides ABCD into 4 triangles (ABN, BCN, CDN, DAN) of equal area.
4) Convex or concave.

Trapezoid (or Trapezium) - any quadrilateral with at least one pair of opposite sides parallel. Properties:
1) Intersection with Cyclic Quadrilateral is an Isosceles Trapezoid.
2) Diagonals divide each other in same ratio.
3) Diagonals intersect on line connecting midpoints of // sides.
4) Sums of two (distinct) pairs adjacent angles equal.
5) In sketch, areas of triangles ADE and BCE are equal.
6) Convex Area = ½(sum // sides) times perpendicular distance between // sides.
7) Convex or crossed.

Kite - any quadrilateral with at least one axis of symmetry through a pair of opposite angles (vertices). Properties:
1) Two distinct pairs of adjacent sides congruent.
2) At least one pair of opposite angles congruent.
3) Diagonals are perpendicular.
4) At least one pair opposite angles bisected by a diagonal.
5) Convex or concave.

Isosceles Trapezoid (or Trapezium) - any quadrilateral with at least one axis of symmetry through a pair of opposite sides. Properties:
1) Two distinct pairs of adjacent angles congruent.
2) At least one pair of opposite sides congruent.
3) Diagonals are equal.
4) Diagonals divide each other into equal parts.
5) At least one pair opposite sides parallel.
6) Convex or crossed.
7) Convex area = ½bsin A*(a + c), in sketch.

Parallelogram (self-dual) - any quadrilateral with half-turn (or point) symmetry. Properties:
1) Opposite sides congruent.
2) Opposite angles congruent.
3) Opposite sides parallel.
4) Diagonals bisect each other.
5) Diagonals divide parallelogram into 4 triangles of equal area.
6) Convex.

Rhombus - any quadrilateral with two axes of symmetry, each through a pair of opposite angles (vertices). Properties:
1) All sides are congruent.

Rectangle - any quadrilateral with two axes of symmetry, each through a pair of opposite sides. Properties:
1) All angles are congruent.

Triangular Kite - any kite with at least three equal angles. (In the sketch, these are at A, B and D). Properties:
1) Tangential points bisect a pair of adjacent sides (AB and AD respectively at Y and Z in the sketch).

Trilateral Trapezoid (or Trapezium) - any isosceles trapezium with at least three equal sides. (In the sketch, these are AB, AD and DC). Properties:
1) Diagonals bisect a pair of adjacent angles (at B and C in the sketch).

Right Kite - any kite inscribed in a circle. Properties:
1) In sketch, ∠B = ∠D = (∠A + ∠C)/2.
2) At least one pair of opposite right angles.

Isosceles Circum Trapezium - any isosceles trapezium circumscribed around a circle. Properties:
1) In sketch, AB = DC = (AD + BC)/2.

Square (self-dual) - any rhombus with an axis of symmetry through a pair of opposite sides or any rectangle with an axis of symmetry through a pair of opposite angles (vertices). Properties:
1) Rotational symmetry of order 4.
2) Respective pairwise intersections of rhombus and rectangle, triangular kite and trilateral trapezium, and right kite and isosceles circum trapezium.

Note: Most of the quadrilateral properties above are proved in my book Some Adventures in Euclidean Geometry, and available as downloadable PDF or printed book. My Key Curriculum Press book Rethinking Proof with Sketchpad also contains some proof activities for an Isosceles Trapezoid, Cyclic Quadrilateral, a Circumscribed Quadrilateral, Rhombus, and the Interior Angle Sum of a Crossed Quadrilateral, and some discussion, and generalizations to higher polygons, in the Teacher Notes.

Michael de Villiers, created, 2008; most recently updated 26 September 2016.