Viviani’s theorem and related problems

 
LITERATURE REVIEW 
Viviani’s theorem is an easily-understood theorem for equilateral triangles. With it having a simple 
condition and significant result, the theorem has gained much attention from mathematicians to further 
study and extend the original statement. To gain a thorough insight of what has been discovered and 
achieved in the light of Viviani’s Theorem, previous work are studied and investigated in this section.
I. 
Original Viviani’s Theorem for Equilateral Triangle: 
Proposed by Vincenzo Viviani, a famous Italian Mathematician, Viviani’s theorem stated: 
 
 
 
 
Theorem 1. For any interior point P of an equilateral triangle ABC, the sum of 
the distances from P to the sides of 
△ABC is constant and equals to the height of 
△ABC: 
 
 
 
s + u + t = h 
 

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2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems 
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Proof 
The core idea of the proof is to find the relationship between the area and 
the height of the triangle. Given 
is an equilateral triangle whose height 
is h and whose side is a. P is an arbitrary point in the given triangle. Let 
, and u be the distances of P to the sides of the triangle. Construct lines 
from to 
, , and to form triangles 
,
and 
whose areas 
are 
,
and  respectively. 
Hence:
+
+
=
Then, we can conclude that: 
+ + = ℎ 
II. 
Extension of Viviani’s Theorem: 
1. Parallelogram
The theorem was extended from the original Viviani’s theorem for equilateral triangle to parallelogram 
(Zhibo Chen, Tian Liang 2006). The extension proposed: