6. Conditions for polygons to possess CVS property Conditions for polygons to possess CVS property (Elias Abboud, 2006), which was found and proven by
linear programming, is stated as:
Proof Given a triangle
△ ABC, let a 1 , a 2 , a 3 be the lengths of the
sides BC, AC, AB respectively. Let P be a point inside the
triangle and let h 1 , h 2 , h 3 be the distances from the point P to the three sides respectively. Let V(P) be the sum of
distances from point P to the sides of the triangle. For 1 ≤ i ≤ 3, let x i = ∑
, where
∑
= V(P) Clearly, for each 1 ≤ i ≤ 3, we have 0 ≤ x i ≤ 1 and ∑
= 1. Denote x = (x 1 , x 2 , x 3 ) and consider the linear function
in three variables F(x) = ∑
. Now, this function is
closely related to the function V. Accurately,
F(x) = ∑
= ∑
∑
= ( )
where S is the area of the triangle.
Consequently, F(x) = ∑
takes equal values in a subset of points of the feasible region if and only if
the function V takes equal values at the corresponding points inside the triangle.
Thus we may define the following linear programming problem; The objective function is:
F(x) = ∑
subject to the following constraints:
∑
≤ 1
≥ 0 ; 1 ≤ ≤ 3
Theorem 7. If V takes equal values at three non-collinear points, inside a convex polygon, then the polygon has the CVS property .
2020 Singapore Mathematics Project Festival Viviani’s Theorem and its Related Problems
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