2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
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ABSTRACT
Out of the many theorems related to geometry, this report will be focusing on the Viviani’s Theorem and
its related problems. The theorem states that the sum of distances from any
point inside an equilateral
triangle is constant and is equal to the triangle’s height. Any polygon that
possesses constant sum of
distances from an arbitrary interior point to the sides is said to have CVS
(constant V(x) sum) property.
The two main objectives of the project are to approach Viviani’s theorem and its extension using vectors
and to establish a relationship linking the results of the earlier approach with Carnot’s theorem - another
theorem dealing with polygons.
From the first
discovery, it was found that in any polygons with CVS property, the sum of unit vectors that
are perpendicular to the respective sides of the polygons is a zero vector. This key finding helped clarify
the rationale behind the result yielded in some earlier studies and helped further investigate some special
geometrical properties to understand why certain polygons have CVS property. In the second discovery,
Carnot’s theorem was further extended to create a clear connection between the two theorems, Carnot’s
and Viviani’s, due to the significant similarity in the two models observed.
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