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 as shown below.
Opposite Side Quadrilateral Properties by Kalogerakis
In 2015, Thanos Kalogerakis from Greece discovered the following interesting properties:
Given an opposite side quadrilateral ABCD with AB = CD and X, Y, M, N the respective midpoints of AD, BC, BD, AC. Construct the circle KMN and locate the point L on the circle directly opposite K. Label as I the intersection of XY with the circle KMN. Draw line IL and label its intersection with AD as U and its intersection with BC as V. Then the follow results hold:
1) XMYN is a rhombus
2) XY is parallel to the angle bisector of AED
3)
4) Line EL bisects ∠AED
5) MN bisects UV.
(Note: While the first two properties are mentioned and proved on pp. 90-91 of my Some Adventures in Euclidean Geometry (free download), the other three results are new.
Exploration
The dynamic sketch below illustrates the above results with some measurements.
Explore the above results dynamically by dragging the red vertices of the quadrilateral.
6) Do the results also hold when ABCD becomes concave or crossed?
7) When will XMYN be a square?
8) What happens when AB // CD? Do the above results still hold?
9) Explore 6-8 by dragging the vertices of ABCD.
Opposite Side Quadrilateral Properties by Kalogerakis
Challenge
10) Can you explain why (prove that) the above properties of an opposite side quadrilateral are true?
Special Case
An interesting special case of property 1) above is that XMN becomes equilateral when ∠B + ∠C = 120o (or ∠A + ∠D = 120o), in which case, XMYN becomes a 120o 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).
Further Challenges
11) Prove that an opposite side quadrilateral that is cyclic (convex or crossed) is an isosceles trapezium
12) 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).
13) 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.
14) 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.
15) Prove or disprove the following statement: An opposite side quadrilateral with a pair of opposite angles equal is a parallelogram.
16) Proofs of 11), 12), 14) are respectively given on pp. 125-126, pp. 173-174 and p. 149, and compare your answer to 15) respectively to pp. 112-113; 128-129 in my Some Adventures in Euclidean Geometry (free download) book.
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.
Another Property
Another interesting property of an opposite side quadrilateral can be found at: click here.
Some Related Links
Another Property of an Opposite Side Quadrilateral
A Property of an Opposite Angle Quadrilateral
Quadrilateral Similar Triangles Collinearity
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
A generalization of Varignon's Theorem
Van Aubel's Theorem and some Generalizations
A diagonal property of a Rhombus constructed from a Rectangle
The 120o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem
Matric Exam Geometry Problem - 1949 (A variation of Reim's theorem)
Golden Rhombus
A Rhombus Angle Trisection Result
A problem by Paul Yiu and its generalization
More Properties of a Bisect-diagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
A Van Aubel like property of an Equidiagonal Quadrilateral
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
A 1999 British Mathematics Olympiad Problem and its dual
Pitot's theorem about a Tangential Quadrilateral
Extangential Quadrilateral
Theorem of Gusić & Mladinić
Constructing a general Bicentric Quadrilateral
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Another generalization of Varignon's theorem
Merry Go Round the Triangle
Japanese Circumscribed Quadrilateral Theorem
Some External Related Links
The original posting of Thanos Kalorakis in 2014 (on a Greek forum)
Equilic Quadrilateral
Quadrilateral (Wikipedia)
Thanos Kalogerakis' Collinearity in Triangle
Thanos Kalogerakis's Problem in Circle and Square
Equilateral Triangle and Semi-circles problem
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Created with WebSketchpad by Michael de Villiers, 6 Dec 2024.