Another Property of an Opposite Side Quadrilateral

In my book Some Adventures in Euclidean Geometry book (free download), first published in 1994, I suggested the name 'side quadrilateral' for a general quadrilateral having at least one pair of opposite sides equal. However, I would now instead like to propose the more appropriate name 'opposite side quadrilateral'.
Perhaps somewhat unexpectedly, an 'opposite side quadrilateral' has some interesting properties one of which shown below.

Another Property of an Opposite Side Quadrilateral
Though several properties of an opposite side quadrilateral are already given at: click here, here is another surprising property:
Given an opposite side quadrilateral ABCD with AB = CD and an isosceles triangle PAD is constructed so that ∠PAD = ∠PDA = ½(∠ABC + ∠BCD). Then triangle PBC is also isosceles and similar to PAD.

Illustration
The dynamic sketch below illustrates the above property with some measurements.
1) Explore the result dynamically by dragging the red vertices of the quadrilateral.
2) Do the results also hold when ABCD becomes concave or crossed?
3) Explore 2) by dragging the vertices so that ABCD becomes concave or crossed. What do you notice?
4) What happens when AB // CD? Do the above results still hold?

Web Sketchpad

Another Property of an Opposite Side Quadrilateral

Challenge
5) Can you explain why (prove that) the above property of an opposite side quadrilateral is true?
6) Compare your proof with the proof on pp. 92-93 of my Some Adventures in Euclidean Geometry (free download) book.

Further Challenges
7) Prove that an opposite side quadrilateral that is cyclic (convex or crossed) is an isosceles trapezium
8) Prove that an opposite side quadrilateral with equal diagonals is an isosceles trapezium. (In the crossed case, however, the pair of equal opposite sides need to cross each other).
9) Prove that an opposite side quadrilateral that has a pair of adjacent angles equal (but the pair of adjacent angles does not include one of the pair of equal opposite sides) is an isosceles trapezium.
10) Prove that the length of one of the equal sides of an opposite side quadrilateral that is tangential (circumcribed to a circle) is the average of the lengths of the other two sides.
11) Prove or disprove the following statement: An opposite side quadrilateral with a pair of opposite angles equal is a parallelogram.
12) Proofs of 7), 8), 10) are respectively given on pp. 125-126, pp. 173-174 and p. 149, and compare your answer to 11) to pp. 112-113; 128-129 in my Some Adventures in Euclidean Geometry (free download) book.

Special Case
An interesting special case of this property is that triangles PAD and PBC become equilateral when ∠B + ∠C = 120^o (or ∠A + ∠D = 120^o), in which case, XMYN becomes a 120^o Rhombus.
Note that a quadrilateral in which a pair of opposite sides have the same length and are inclined at 60 degrees to each other as in this special case has been called an Equilic Quadrilateral by Garfunkel (1981) and Honsberger (1985).

References
De Villiers, M. (2009). Some Adventures in Euclidean Geometry book (free download), Lulu Publishers: Dynamic Mathematics Learning.
Garfunkel, J. "The Equilic Quadrilateral." Pi Mu Epsilon J. 7, 317-329, 1981.
Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 32-35, 1985.

Some External Related Links
Equilic Quadrilateral
Quadrilateral (Wikipedia)

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Created with WebSketchpad by Michael de Villiers, 7 Dec 2024.