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On Quadrilaterals
Let Them Explore and Guess
Michael Keyton
IMSA
[email protected]
On Quadrilaterals
In high school geometry courses, the study of quadrilaterals is limited. Usually, only six
(sometimes at many as eight) quadrilaterals are defined and studied peripherally. The six studied
are the trapezoid, isosceles trapezoid, parallelogram, rhombus, rectangle, and square (sometimes
kite and cyclic quadrilateral). We will look at many more, discuss definitions, and make
suggestions for students to make the study of geometry more mathematical.
On many occasions, I have argued and supported inclusive definitions for each of these.
Historically, they were all given exclusively, but as time passed all now have inclusive
relationships except for the trapezoid and isosceles trapezoid. Typical definitions for the
quadrilaterals are:
Trapezoid - a quadrilateral with exactly one pair of parallel sides (I will argue to change this
definition and supply reasons for why it has not been universally done)
Isosceles trapezoid – a trapezoid with congruent non-parallel sides (again suggestions for
change)
Parallelogram – a quadrilateral with both pairs of opposite sides parallel
Rectangle – a parallelogram with a right angle (this has been changed over the years, to be
discussed later)
Rhombus – a parallelogram with congruent consecutive sides
Square – a rectangle with congruent consecutive sides.
In Euclid, rectangles and rhombus were not allowed to be parallelograms, squares were not
allowed to be rectangles or rhombi. All definitions of quadrilaterals were exclusive.
Part I – Inclusive Definitions and Definitions
Inclusive definitions are much more powerful mathematically than exclusive ones. Having
definitions for the rectangle such as above serves three outstanding mathematical purposes –
first, if A is defined to be a subclass of B, then any theorem proved for B becomes automatically
true for A; second, this gives more new theorems; and third, a converse does not require an or
conclusion. For example, with exclusive definitions such as above, the following theorem causes
difficulty. If the diagonals of a quadrilateral separate each other proportionally, then the
quadrilateral is a trapezoid or a parallelogram. So why do we still have a few remnants of
exclusive definitions. I believe because in high school geometry courses, there are very few
theorems for the quadrilaterals. For the trapezoid there is usually only two theorems given: (1)
The median of a trapezoid is the arithmetic average of the parallel sides. (2) the area of a
trapezoid is one half the product of the height and the sum of the bases.
In an earlier paper, I gave a list of theorems about trapezoids.
Definitions are arbitrary. They can be changed, but there are characteristics that make some
better than others. A good definition should have the following properties:
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(1) It should be define the new object using known terms (previously defined words or undefined
words).
(2) It should be minimal, requiring as few parts to verify as possible,
(3) it must be reversible,
(4) it should distinguish the object being defined from other objects
(5) it should be inclusive.
`When I use a word,' Humpty Dumpty said, in rather a scornful tone, `it means just what I choose
it to mean -- neither more nor less.' Lewis Carroll, Through the Looking Glass
An example of a poor definition for a rectangle is a quadrilateral with four right angles. This
violates the minimal condition, for to verify that a quadrilateral is a rectangle requires four tests,
when only three are needed. A better definition is a rectangle is a quadrilateral with three right
angles. This allows a theorem – if a quadrilateral has three right angles, then it has four right
angles.
If a theorem and its converse about a quadrilateral X are both true, then the definition of the
quadrilateral can be replaced with the theorem. Thus, we see the following results and converses
about parallelograms:
(1) a quadrilateral with both pairs of opposite sides parallel
(2) a quadrilateral with both pairs of opposite sides congruent
(3) a quadrilateral with diagonals that bisect each other
(4) a quadrilateral with both pairs of opposite angles congruent
(5) a quadrilateral with a pair of opposite sides congruent and parallel
(6) a quadrilateral with all a pair of opposite angles that are congruent and supplementary to one
of the other angles
There are many more that could be stated. Which one is the best definition for a parallelogram?
The answer is any. But (1) conveys the linguistic meaning of the word "parallelogram." For the
others, should that be thought of as quadrilaterals (Four sided figures) or quadrangles (four
angled figures), or just figures.
In alignment with these characteristics, I offer the following as definitions of the quadrilaterals.
A cyclic quadrilateral is a quadrilateral with vertices on a circle.
A trapezoid is a quadrilateral with a pair of parallel sides.
A parallelogram is a quadrilateral with two pairs of parallel opposite sides.
A rectangle is a quadrilateral with 3 right angles.
An isosceles trapezoid is a quadrilateral with two disjoint pairs of consecutive congruent angles.
A kite is a quadrilateral with two disjoint pairs of consecutive congruent sides.
A rhombus is a quadrilateral with congruent sides.
A square is a quadrilateral with congruent sides and a right angle.
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Notice, the complexity of the figure depends upon the number of conditions that it needs to
satisfy. A trapezoid and the cyclic quadrilateral need only 1 condition; parallelograms, kites, and
isosceles trapezoids need two; rectangles and rhombi need three; and the square needs four.
Part II Other quadrilaterals
Why are we restricted to just these quadrilaterals? I offer the following as some of the
reasons.
The square is the basic unit for measuring area. It also has the maximum symmetry.
The rectangle is an immediate generalization of the square for area.
The parallelogram allows us to transfer angles and distances in the plane.
The rhombus transfers distances in two directions. It is the intersection of two congruent circles.
The trapezoid is an easy way to measure areas. (not in Euclid)
The kite is formed by the intersection of two circles. It also has a line of symmetry. (not in
Euclid)
The isosceles trapezoid has symmetry.
The cyclic quadrilateral transfers angles through space.
Are there others? Since Euclid, we have added the cyclic quadrilateral, trapezoid, and kite.
Others which appear some in advanced studies are:
(a) An orthodiagonalquadrilateral is a quadrilateral with perpendicular diagonals.
(b) An isodiagonalquadrilateral is a quadrilateral with congruent diagonals.
Number of characteristics:
(1) A ___________ is a quadrilateral with a right angle.
A ____________ is a quadrilateral in which one diagonal bisects the other.
(2) A ____________ is a quadrilateral with perpendicular opposite sides.
(2) A ____________ is a quadrilateral with perpendicular and congruent diagonals.
The objective is not only to define but also to find theorems about each of the new quadrilaterals.
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III Other investigations of existing quadrilaterals.
The investigations in the books are about the angles, the sides, and the diagonals. How about
including (a) the intersection point of the diagonals (the diacenter), (b) the medians (segments
from midpoints of opposite sides, (c) the intersection point of the medians (medcenter), (d) the
segments from midpoints of adjacent sides (???), (e) segments from the vertices to a midpoint of
an opposite side, (f) quadrilaterals formed by the intersections of segments in (e), (g) altitudes
from vertices to opposite sides (note there are 8), (h) quadrilaterals formed by the intersections of
the lines in (g), (i) angle bisectors, (j) quadrilaterals formed by intersections of lines in (i), (k) the
centroids, circumcenters, incenters, orthocenters of the 8 triangles formed by the diagonals and
the sides of the quadrilateral, (l) the quadrilaterals formed by the intersections of the points in
(k), (m) parallels through the vertices to the diagonals, (n) perpendiculars to the diagonals from
the vertices of the quadrilateral,
In general if we think of a property X and a quadrilateral Z, we can form two theorems:
If Z has property X, then the result is ______________
In the article “Students Discovering Geometry using Dynamic Geometry Software” in the
book Geometry Turned On (MAA, 1997), I give ideas for about 200 different explorations that
students can pursue on quadrilaterals. In the years I have used explorations using software, I
have seen more than 1000 new results from students. Some other papers can be found at
www.imsa.edu/~keyton in the geometry section.
The main results that I have found with and for students is that they begin to appreciate the
necessity of proof, the difficulty with creating definitions and stating conjectures, the pleasures
and the agony in the discovery methods, seeing how immense geometry is, and demonstrating
their creativity beyond what I could ever conceive.
Never overestimate what a student knows;
never underestimate what he can produce.
Michael Keyton
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