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Exotic Quadrilaterals Tim Craine, CCSU (retired) ATOMIC Conference, December 4, 2017 A Hierarchy of Familiar Convex Quadrilaterals Circumcentric and Incentric Quadrilaterals

• A quadrilateral is said to be cyclic (or “circumcentric”) if it has a circumscribed ; that is, all four vertices lie on the same circle.

• A quadrilateral is said to be tangential (or “incentric”) if it has an inscribed circle; that is, all four sides are to the same circle. Properties of Circumcentric and Incentric Quadrilaterals

• Opposite of a circumcentric quadrilateral are supplementary; that is the sum of their measures is 180°. • The sum of the lengths of one pair opposite sides of an incentric quadrilateral is equal to the sum of the lengths of the other pair of opposite side. • Conversely, if opposite angles of a quadrilateral are supplementary, then it is circumcentric; and if the sum of the lengths of pairs of opposite sides are equal, then it is incentric. Potential Properties of Sets of Quadrilaterals

Perpendicular Adjacent Opposite Adjacent Congruent Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of

S R Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals O Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of Kites (1st orientation)

Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of Kites (2nd orientation)

Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of Isotraps (1st orientation)

Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of Isotraps (2nd orientation)

Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of

Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

Properties of

Perpendicular Adjacent Diagonal Opposite Adjacent Congruent Diagonals Sides Bisects Sides Angles Diagonals Congruent Diagonal Parallel Congruent SP = PQ OQ = OS PQ || RS m

PR ┴ QS QR = RS m

PR bis

What other classes of quadrilaterals exist?

Level 1: Any one of the 12 properties listed in the chart defines a class of quadrilaterals. • Some already have names. For example, ✓ A is a quadrilateral with at least one pair of parallel sides ✓ A circumcentric quadrilateral is one that has a circumscribed circle. ✓ An incentric quadrilateral is one that has an inscribed circle. • Others may need new names. For example ✓ We could call a quadrilateral with one at least one pair of opposite sides congruent “trans-equilateral” and a quadrilateral with at least one pair of adjacent sides congruent “cis-equilateral.” Thus parallelograms and isotraps are special cases of trans-equilateral quadrilaterals whereas parallelograms and kites of special cases of “cis-equilateral” quadrilaterals. What other classes of quadrilaterals exist?

Level 2: Pick two properties and see is they define a new class of quadrilateral Caution: If both properties belong to one of the five highlighted Level 2 quadrilaterals*, chances are you will not find anything new. Exceptions will be discussed later. *, (1st orientation), Kite (2nd orientation), Isotrap (1st orientation) and Isotrap (2nd orientation). For example: If a quadrilateral has one pair of opposite angles congruent and one pair of opposite sides congruent, then it is a parallelogram. What other classes of quadrilaterals exist?

Level 2 (continued) A good strategy is to two properties belonging to kites or isotraps, one from each orientation. Example: a quadrilateral with two pair pairs of adjacent sides congruent, one from each orientation of the kite (i.e. two overlapping pairs of congruent adjacent side). See the “tri-equilateral” quadrilateral below. Another strategy is to pick one property belonging to kites and one property belonging to isotraps. Example: a quadrilateral with diagonals that are both perpendicular and congruent. See the “orthodiagonal equidiagonal” quadrilateral below. What other classes of quadrilaterals exist?

For each new class of quadrilateral, look for possible sub-classes by asking these questions:

Is it possible for this quadrilateral to be: a kite (that is not a )? a parallelogram (that is neither a rhombus nor a )? an isotrap (that is not a rectangle)? a rhombus (that is not a )? a rectangle (that is not a square)? In answering these questions you may discover a new class of Level 3 quadrilaterals. Some examples of new classes of quadrilaterals

• Experiment by constructing quadrilaterals with 2 properties (not from the same level 2 class). Use a GeoGebra sketch or other tools • Answer the questions on the previous slide. Tri-equilateral quadrilateral Tri-equiangular quadrilateral Equidiagonal-Orthodiagonal Quadrilateral Circumcentric-Orthodiagonal Quadrilateral Incentric-Equidiagonal Quadrilateral www.revolvy.com/main/index.php?s=Bicentric quadrilateral Skew Kite Skew Isotrap SSA Lemma

If two sides of one are congruent to two sides of another triangle and if the angles opposite one pair of sides are congruent to each other, then the angles opposite the other pair of sides are either congruent or supplementary.

sin C = sin C’ = sin C” = (c/b) sin B; C’ + C” = 180° Ambiguous Kite—Circumcentric Quadrilateral Ambiguous Parallelogram-Isotrap Ambiguous Parallelogram-Kite Another ambiguous isotrap? Making room for more quadrilaterals in the hierarchy