Viviani’s theorem and related problems

CONCLUSION 
In conclusion, the two main objectives of the project is to approach Viviani’s theorem and its extension 
using vectors and to introduce a connection inspired from the results of the earlier approach with another 
theorem in the same field, Geometry in Mathematics.
Overall, through a new way of approach towards the original Viviani’s theorem and its extension, it is 
discovered that polygons that possess CVS property would have the sum of the unit vectors, such that each 
of them is perpendicular to its corresponding side, is a zero vector. This key finding of the project explained 
the rationale behind the result yielded of Elias Abboud, further investigated some special geometrical 
properties of Viviani polygons, thus, explained why some certain special polygons, such as regular 
polygons, parallelograms, polygons with opposite sides parallel would be CVS polygons.
Carnot’s theorem, whose model is found to have a similarity with Viviani’s theorem, is further studied and 
extended so that the link between those two theorems would be established, making the relationship 
between two theorems clearer. As the results, two problems (Link 1 and Link 2) have been introduced with 
the suggested solution merely developed from the all the earlier work of the project, especially the two 
corollaries specifically. However, there remains a challenge which is unsolved due to time constraint, which 
is believed to make the project be more inclusive and successful if it is completely worked out.
In future work, the project could be further investigated in three-dimension since vectors are a strong tool 
in Geometry. The Corollaries found could hold true even when generalized from 2D to 3D. The results, if 
held true in both 2D and 3D, would leave a larger impact and be more significant compared to the results 
that only hold true in two-dimensional space.
 
 

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2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems 
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