Kimberly Burrell, Brad Simmons, and Doug Westmoreland
Pappus of Alexandria was a
Greek Mathematician. In 320 A.D. he composed a work with the title Collection
(Synagoge). This work was very important because on several reasons.
It is the most valuable historical record of Greek Mathematics that would
otherwise be unknown to us. We are able to learn that Archimedes'
discovered the 13 semiregular polyhedra, which are today known as
"Archimedian solids." He also include alternate proofs and
supplementary lemmas for propositions from Euclid, Archimedes, Apollonius, and
Ptolemy. Pappus' treatise includes new discoveries and generalizations not
found in early work. The Collection contained eight books.
The first book and the beginning to book two have been lost. In Book IV,
Pappus included an elementary generalization of the Pythagorean theorem.
He also included the following problem, which has came to be known as the Pappus
areas theorem. It is not known whether or not the problem originated with
Pappus, but it has been suggested that possibly it was known earlier to Heron.
Consider any triangle ABC.
Construct the external parallelogram
on side AB.
Construct the external parallelogram
on side AC.
Extend the external parallels
to intersect at point P and draw segment AD.
Now construct the external
parallelogram on side BC.
Now we can prove that the
sum of the parallelogram areas on sides AB and AC equals the area
of the parallelogram on side BC. This is what is known as Pappus
of the Pappus Area Theorem
In the following sketch,
one can notice that the sides of the parallelogram drawn on side
BC are used to construct lines that are parallel to segment PA.
Now we can consider these triangles PEA and JDB, and
these triangles are congruent by AAS (angle-angle-side) congruency
Angle APE and Angle BJD
are congruent because they are corresponding angles on a pair
of parallel lines cut by a transversal, which is line GP.
Angle PEA and angle JDB
are congruent also because they are corresponding angles on a
pair of parallel lines cut by a transversal, which is line DP.
Side EA and side DB are
congruent because they are the opposite sides of a parallelogram.
By a similar approach one
can see that triangle PAF is congruent to triangle KCG.
Now by use of substitution,
parallelogram PABJ is congruent to parallelogram EABD and parallelogram
PACK is congruent to parallelogram FACG. The sketch below
indicates this relationship and shows the replacement by substitution.
If we extend segment
DA by constructing a line, then we will be able to construct a
line through point H that is parallel to AB and a line through
point I that is parallel to AC.
Angle ABC is congruent to
angle LHI due to the fact that they are corresponding angles on
a pair of parallel lines cut by a transversal, which is line HJ.
Angle ACB is congruent to angle LIH due to the fact that they
are corresponding angles on a pair of parallel lines cut by a
transversal, which is line IK. We know that BC is congruent
to HI since opposite sides of a parallelogram are congruent.
From this information triangle
ABC is congruent to triangle LHI by ASA (angle-side-angle) congruency
postulate. The sketch below shows this relationship.
Using substitution once
again, we replace triangle ABC with triangle LHI. This forms
the hexagon BACILP that has the same area as parallelogram BCIH.
The sketch below shows this
The corresponding angle
of parallelogram JPAB and parallelogram BALH are congruent.
In this side AB is congruent to side JP and side HL. In
the original construction, the length of side BH is congruent
to segment PA. Hence, PA is congruent to AL.
Given the above, parallelogram
JPAB is congruent to parallelogram BALH and parallelogram KPAC
is congruent to CALI.
Thus, hexagon JPKCAB is
congruent to hexagon BACILH.
Since substitution was used
we can retrace the steps and conclude that the sums of the areas
of parallelograms EDBA and FGCA are equal to the area of parallelogram
Click here for an animated GSP sketch.