Viviani’s theorem and related problems

 
METHODOLOGY 
At the very start, when sourcing for a field of Mathematics to investigate, we looked into previous year’s 
published submissions to get a rough idea of how extended our projects should be. Among different areas 
of Math, we found Geometry to be the least complicated to approach and extend as most of the theorems 
and problems can be visualised with figures and they hardly have fixed solutions.
Next, we searched the Internet, specifically sites such as Wikipedia.com, cut-the-knot.org, xmltwo.ibo.org, 
for papers, reports and articles providing deeper insights into some problems. After putting all problems 
and theorems we felt interested in into a spreadsheet, we compared them using common factors like: 

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2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems 
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availability of previous work, applicability, etc. We narrowed our choices down to theorems related to 
triangles, because triangles are the most basic polygons (3-sided polygons), which promises further 
extension by altering conditions, such as increasing the number of sides or introducing new lines.
Finally, we decided that Viviani’s Theorem would be the topic of our project on the grounds that it features 
basic constructions that are easy to follow, that its algebraic components can be recorded and observed 
using basic programming on software like Microsoft Excel, and that it inspires us with different ways to 
extend.
During our project, we used the online application, GeoGebra to confirm the conditions and requirements 
needed for our extension and proof of the Viviani’s Theorem, as well as to create shapes used in our report 
and presentation. We constructed the figure based on the original conditions of the theorem, then tried 
dragging points around and adding/eliminating points and lines. To examine the CVS property of each 
polygon, we wrote a sum function of all the distances and linked it to an Excel sheet, which automatically 
recorded the sums as we changed the position of the arbitrary interior point. This helps us a lot in testing 
our hypothesis and spotting the exceptions.