2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
5
P
+
+
+
=
ℎ
× (
+
)
Since the left-hand-side expression is a constant, the sum of distance is aso a constant. Thus, it is
independent of the location of point
P.
Proof without words using vectors (Hans Samelson, 2003)
| | = | | = | |
+
+
= 0
. + . + .
= 0
( − ′) + ( − ′) + ( − ′) = 0
+ + = ′ + ′ + ′
The converse also holds:
2.
2k-sided polygons with parallel opposite sides
From the result above on the parallelogram, it is generalized to 2
n-gon with opposite sides parallel. In other
words, it is stated:
Let
P be the arbitrary interior point in the polygon. Let the
opposite parallel side of
be
.
Since the polygon has
2k sides, which is an even number
and opposite sites are parallel, those pairs of opposite sides
parallel would also have the same length.
Let the length of the first pair of opposite sides parallel be
.
Theorem 2.2. If the sum of the distances from a point in the interior of
a quadrilateral to the sides
is independent of the location of the point, then the
quadrilateral is a parallelogram
.
Theorem 3. In a 2k-sided polygon with opposite sides parallel, the sum of
distances from an arbitrary point inside the polygon to its sides is constant and is
independent of the point’s position.
2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
6
(
)
(
)
+
=
(
+
)
Noted that the sum of all the triangles with the same vertex
P equals to the area of the polygon and
+
is indeed the distance between the pair of parallel sides. Since the sum of distances between any pair
of opposite
parallel sides is constant, it follows that the sum of all pairwise
sums between the pairs of
parallel sides, is also constant. However, the converse of this generalization is not true. A counterexample
that can be easily found is an
equilateral hexagon, which does not necessarily have opposite sides parallel
but still have constant sum of distances from an arbitrary point.
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