2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
7
The Viviani’s theorem holds not only for regular polygons but also for equiangular polygons. The
extension, when reducing the condition of the polygon from “regular” to “equiangular” becomes more
relaxed compared to the previous results. It states
:
In “On Viviani’s Theorem and its Extensions” (p11, Example 4.6), Elias Abboud describes a proof for the
extension above by locating inside the equiangular polygon a regular
n-gon using parallel lines constructed.
Here, the proof is represented.
Proof 1 (Elias Abboud. 2009)
Let equiangular
n-gon
ABCDE be
ℋ.
Locate
inside
ℋ a regular
n-gon, ℱ. Rotate ℱ
around its centroid until one
of its sides is parallel
to one side of
ℋ. As both polygons have the same
number of sides, the included angle in-between any
two adjacent sides in each polygon is fixed. Hence,
when one side of
ℱ is parallel to one side of ℋ, all
corresponding sides of both polygons are parallel.
Let
ℋ
be the function of the sum of distance from
point
O to sides of polygon
ℋ and
ℱ
be the function
of the sum of distance from point
ℱ
to
sides of
polygon
ℱ
:
ℋ
=
ℱ
+ FH
1
+ GH
2
+ IH
3
Because the area and lengths of the two polygons are fixed by condition,
distances between its
corresponding sides
FH
1
, GH
2
, IH
3
are constant. Thus, for any point
O inside
ℱ we have,
ℋ
=
ℱ
+ c,
where
c represents the sum of distances between the parallel sides of
ℋ and ℱ.
By Viviani’s theorem for regular polygon,
ℱ
is constant. Hence,
ℋ
is constant.
(Q.E.D)
However, the converse does not hold true, and a counterexample
is the parallelogram, which has CVS
property though its angles are not equal.
The proof is creative as it tries to reduce the problem to existed problem by construction of perpendicular
and parallel lines. However, the drawback of it is that the construction may take time if done manually.
Beside Elias Abboud,
another mathematician, Michel Cabart has attempted to prove this theorem using
another way. With vector approach, his solution was without words, simply by constructing vectors of Hans
Samelson (Literature Review 1)
Theorem 5. The sum of distances from a point to the side lines of an equiangular
polygon does not depend on the point and is that polygon's invariant
.
2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
8
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