An interesting quadrangle problem

This problem appears in Rethinking Proof with the Geometers Sketchpad, by Michael de Villiers.    It provides a good example of how programs like Maple and Geometers Sketchpad can be used to experiment and conjecture, then even assist with the calculations needed to establish the conjecture (if it is true).

Problem: Given a convex quadrangle [Maple Math], with the vertices labelled clockwise, let [Maple Math]denote the clockwise interior quadrangleof Q, that is the (necessarily convex) quadrangle whose vertices are the intersections of the four 'clockwise' medians of Q, [Maple Math], where [Maple Math]is the midpoint of the segment [Maple Math]for i = 1, 2 , [Maple Math]is the midpoint of [Maple Math], and [Maple Math]is the midpoint of [Maple Math].
Determine bounds on the ratio [Maple Math]

Here is a picture of a typical Q with its clockwise interior quadrangle drawn in blue. The ratio is calculated and displayed below the picture.

[Maple Plot]


Solution[Maple Math][Maple Math][Maple Math]and the bounds are sharp.

Without loss of generality, we can assume that [Maple Math][Maple Math][Maple Math]and [Maple Math]where a and b are positive and a+b > 1. This is because the quadrangle can be so situated with an affine map, which preserve ratios of areas.

Using determinants (see the Maple computations below), the ratio of the areas works out to [Maple Math], where

top = [Maple Math]

and bottom = [Maple Math].

To establish the right hand inequality [Maple Math],we need to show [Maple Math]is never negative. But when factored (see the Maple computations below), this quantity is

[Maple Math]

This is always >= 0 with equality when [Maple Math]for 0 < a < 3 or [Maple Math]for [Maple Math].

Hence the right hand inequality is sharp, in the sense that equality is achieved and no smaller upper bound is possible.

Here is a picture showing the segment [Maple Math]for 0 < a < 3 and the ray [Maple Math]for [Maple Math]

[Maple Plot]

In the case [Maple Math](i. e., (a,b) is on the blue ray with positive slope), we can derive that [Maple Math]which says that the sides of the interior quadrangle with positive slope are parallel. In the case [Maple Math], (i.e., (a,b) is on the blue segment with negative slope), we can derive that [Maple Math], which says that the sides of the interior quadrangle with negative slope are parallel. Hence the ratio is 1/5 when the interior quadrangle is a trapezoid.

To establish the left hand inequality [Maple Math][Maple Math], we need to show that [Maple Math]is always positive. But when factored (see the Maple computations below), this quantity is

[Maple Math]

The factor [Maple Math]is always positive for a > 0, b > 0, and [Maple Math]: To see this,

[Maple Math][Maple Math]

[Maple Math]

[Maple Math]

[Maple Math]

[Maple Math][Maple Math]> 0 except when b = 2

But in that case (b = 2), [Maple Math][Maple Math][Maple Math]> 0

Note that when b = 1, a = 0, the quadrangle collapses to a triangle and the factor is 0.

The factor [Maple Math]is also always positive for a > 0, b > 0, 1 < a+b:

To see this,

[Maple Math][Maple Math]

[Maple Math][Maple Math]> 0.

When a = 1 and b = 0, we get the other degenerate case where the ratio becomes 1/6. The left hand inequality is sharp also, in the sense that there are quadrangles whose area ratio is as close to 1/6 as you want, but the ratio 1/6 is never achieved for a quadrangle.

Since affine maps preserve parallel lines, the 'parallel property' established above is preserved when the normalized quadrangle is 'un-normalized' back to the original, so we have a theorem.

Theorem:The clockwise interior quadrangle is 1/5 of its exterior quadrangle precisely when it is a trapezoid, otherwise it is less than a 1/5 but greater than a 1/6.

Here is an animation which illustratest the theorem.

[Maple Plot]

Note: The same theorem holds if clockwise is replaced by 'counterclockwise' in the above theorem.

Analogous Problems:

1. Ask the analogous question about convex pentagons (hexagons, etc)

2. Let DEF be the triangle interior to ABC formed by joining A to 1/3 B + 2/3 C, B to 1/3 C + 2/3 A, and C to 1/3 A + 2/3 B. Find bounds on area(DEF)/area(ABC).

Maple computations