Theorem 1.3: What can we tell about the geometrical property of polygons that possess CVS
property?
Given a convex polygon. Construct unit vectors from an interior point perpendicularly directed to the sides
of the polygon. If ∑
⃗ = 0⃗, then the polygon has CVS property.
Cases for an n-sided polygon to have ∑
⃗ = 0⃗,
If the number of sides of polygon is odd,
o
The polygon is regular
If the number of sides of polygon is even,
o
The polygon is regular
o
There must be pairs of vectors that are in opposite direction
The polygon has opposite sides that are parallel, since the opposite sides share the same
perpendicular from the given interior point
2. Corollaries
Let
⃗ ∙
⃗ =
⃗ ∙ ( ⃗ +
⃗)
From the equation above,
⃗ ∙
⃗ =
∙
∙ cos (90 −
360
)°
⃗ ∙
⃗ =
∙
∙ cos (90 −
360
)°
…
Corollary 1. Given a regular n-sided polygon with each side having a length of a. Let
,
,
…
be the foot of the perpendicular lines from a random interior point P to
the sides of the polygon, which we defined as
,
, …
. O is another
random interior point. The following equation holds:
⃗ ∙
⃗ =
⃗ ∙
⃗ =
×
(
−
) ∘
2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
19
…
⃗ ∙
⃗ =
∙
∙ cos (90 −
360
)°
Hence,
⃗ ∙
⃗ = ∙
∙ cos (90 −
360
) ∘
On the other hand:
= 2
Thus, Corollary 1 is true.
Given a n-sided convex polygon, it holds:
⃗ ∙
⃗ =
∙
∙ cos (∠
− 90)°
⃗ ∙
⃗ =
∙
∙ cos (∠
− 90)°
…
⃗ ∙
⃗ =
∙
∙ cos (∠
− 90)°
∑
⃗.
⃗ = . ∑
. cos(∠
− 90)°
=
2.
ℎ
.
(90 −
)
Area of equilateral convex polygon = a. ∑
∑
⃗.
⃗ = . ∑
. cos(∠
− 90)°= a. ∑
.
(90 −
)
∑
. cos(∠
− 90)° = ∑
.
(90 −
) (*)
In order to prove that the polygon is regular (both equilateral and equiangular), from (*) we have to prove
that:
∠
= ∠
=. . . . = ∠
Corollary 2. (hypothesis) Given an equilateral polygon. P and O are arbitrary points
inside the given polygon. From P, construct perpendicular lines to sides of polygon.
Let
,
,
, … be the intersections of the constructed perpendicular lines to sides
,
,
, . ..The given polygon is regular if:
⃗ ∙
⃗ =
⃗ ∙
⃗ =
×
(
−
) ∘
2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
20
Dostları ilə paylaş: |