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:
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