2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
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In the field of geometry, triangles are intriguing shapes thanks to their being an important starting point to
explore polygons. Among various studies and theorems related to triangles, Viviani’s Theorem is a study
we find potential as it features basic conditions, thus, there would be opportunities for us to further study
and investigate.
Viviani’s Theorem is established by Italian mathematician Vincenzo Viviani (1622 – 1703). It states:
“In
an equilateral triangle, the sum of distance from an arbitrary point to the three sides is always constant,
and equal to the height of the triangle.”
The theorem has been proven, explored and extended in several ways: proof of converse theorem, extension
to general triangles, extension to equilateral polygons, extension to polyhedra, etc, via numerous methods:
area of triangle, coordinate geometry, linear programming and analysis geometry. However, so far we have
only encountered two studies using the approach of vectors, one being a proof by Hans Samelson (2003)
that generalises the theorem to equiangular polygons, and the other being a proof for converse theorem by
Zhibo Chen and Tian Liang Chen (2006).
Vectors are
a powerful tool in Mathematics, especially in Geometry because
it works with both two-
dimensional and three-dimensional space and can be algebraized despite being a geometrical tool.
In this report, a distance-sum function
V(x) is defined as the sum of distance from interior arbitrary point
P
in the given polygon to sides of the polygon. A polygon is said to possess CVS (constant
V(x) sum) property
if the polygon has constant sum of distance from an arbitrary interior point to its sides.
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