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In plane geometry, a quadrilateral is a polygon with four edges or sides and four vertices.
Sometimes for etymological symmetry with triangle the term quadrangle is used and sometimes
tetragon for consistency with 5 sided pentagon, 6 sided hexagons and so on...
Quadrilaterals are either simple, i.e. not self-intersecting, or complex i.e. self-intersecting. Simple
quadrilaterals are either convex or concave.
Convex quadrilaterals are further classified as follows:
Trapezium / trapezoid: Only one pair of sides is parallel.
? Isosceles trapezium / Isosceles trapezoid: Two opposite sides are parallel, the two other
sides are of equal length and the two ends of each parallel side have equal angles. This
implies that the diagonals are of equal
length.
Parallelogram: both pairs of opposite sides are parallel. This means that opposite sides are
of equal length, opposite angles are equal and the diagonals bisecting each other.
? Mid-point Theorem: If you join the mid-points of all the sides of the parallelogram then you
will get a parallelogram.
Kite: two adjacent sides are of equal length and the other two sides also of equal length. This
implies that one set of opposite angles is equal, and that one diagonal perpendicularly
bisects the other.
Rhombus or rhomb: All four sides are of equal length. This implies that opposite sides are
parallel, opposite angles are equal, and the diagonals
perpendicularly bisect each other.
Rhomboid: A parallelogram in which adjacent sides are of unequal lengths and angles are
oblique (not right angles).
Rectangle (or Oblong): All four angles are right angles. This implies that opposite sides are
parallel and of equal length, and the diagonals
bisect each other and are equal in length.
Square (regular quadrilateral): All four sides are of equal length (equilateral), and all four
angles are equal (equiangular), with each angle a right angle. This implies that opposite
sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect
each other and are of equal length. A quadrilateral is a square if and only if it is both a
rhombus and a rectangle.
Cyclic quadrilateral: The four vertices lie on a circumscribed circle.
Tangential quadrilateral: The four edges are tangential to an inscribed circle. Another term for
a tangential polygon is inscrutable.
Bicentric quadrilateral: Both cyclic and tangential.
More quadrilaterals:
An arrowhead has bilateral symmetry like a kite, but the top concaves inwards.
A self-intersecting quadrilateral is called variously a cross-quadrilateral, butterfly quadrilateral
or bow-tie quadrilateral.
An equiangular quadrilateral is a rectangle if convex, and an "angular eight" with corners on a
rectangle if non-convex.
A quadrilateral whose vertices do not all lie in a flat plane is a skew quadrilateral.
A circle is all points equidistant from a given center point that lie in a same plane.
The distance from the center point to a point on the circle is called the radius of the circle, shown in
the diagram as r. The radius is a line segment with one endpoint on the circle and the other at the
center of the circle.
A diameter has both endpoints on the circle and must also contain the center of the circle. A
diameter is twice the length of the radius.
Segments with both points on the circle are called chords. A diameter is a special type of chord,
which passes through the center of the circle. As with any segments, two chords are congruent if
their lengths are equal.
A portion of a circle is called an arc. Arcs are measured in degrees, like angles, and are classified
in a similar way. There are minor arcs, major arcs, and semicircles. A minor arc measures
between 00 and 1800 . A major arc is between
1800 and 3600 . The semicircle is exactly 1800 .
Two circles that have the same center are called concentric circles
When diameters intersect at the center of a circle, they form central angles. The measure of a
central angle equals the measure of the arc that it "intercepts".
An inscribed angle is an angle with its vertex on the circle and its sides containing chords of the
circle. Angle RST is an inscribed angle.
Chords that intersect at any interior point form "interior angles." Angles r, s, t, and u are interior
angles.
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