Midpoint trapezium theorem
A well known theorem for a trapezium that appears in many school textbooks is the following: Given any (convex) trapezium ABCD with AD // BC and E and F the respective midpoints of opposite sides AB and CD. Then EF = (AD + BC)/2.
Click here for a dynamic sketch illustrating this theorem.
Investigate a Quadrilateral
But what happens in the case of a general quadrilateral?
Below is given a general quadrilateral ABCD with E and F the respective midpoints of opposite sides AB and CD, and measurements given for EF and (AD + BC)/2.
1) Drag any of the vertices of ABCD and carefully compare the measurements of EF and (AD + BC)/2. What do you notice? What conjecture can you make?
2) Ensure that you check your conjecture carefully, and also look at concave and crossed cases as well.
A trapezium theorem generalized
Formulate
3) Check the formulation of your conjecture in 1)-3) above by clicking here.
Challenge
4) Can you explain why (prove that) your conjecture is true?
5) Check or compare your proof in 4) above with those given in the links here.
Investigate Further
5) Click on the 'Link to midpoint segment hexagon' button on the bottom right to navigate to a new sketch showing a dynamic hexagon with G and H the respective midpoints of the opposite sides AB and DE of the hexagon ABCDEF. Measurements for GH and (AF + FE + BC + CD)/2 are given.
6) Drag any of the vertices of ABCDEF and carefully compare the measurements of GH and (AF + FE + BC + CD)/2. What do you notice? What conjecture can you make? Ensure that you check your conjecture carefully, and also look at concave and crossed cases as well.
7) Can you use the first result for a quadrilateral to explain why (prove that) your hexagon conjecture is true? Can you generalize further to an octagon or decagon? Check and compare your explanation (proof) by clicking here.
Related Links
Midpoint Trapezium Theorem
Some Trapezoid (Trapezium) Explorations
Visually Introducing & Classifying a Trapezoid/Trapezium (Grades 1-7)
Matric Exam Geometry Problem - 1949
Tiling with a Trilateral Trapezium and Penrose Tiles (PDF)
Some Properties of Bicentric Isosceles Trapezia & Kites
Visually Introducing & Classifying Quadrilaterals by Dragging
Introducing, Classifying, Exploring, Constructing & Defining Quadrilaterals
A Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
The Parallel-pentagon and the Golden Ratio
International Mathematical Talent Search (IMTS) Problem Generalized
Clough's Theorem (a variation of Viviani) and some Generalizations
A Geometric Paradox Explained
Semi-regular Angle-gons and Side-gons: Generalizations of rectangles and rhombi
Intersecting Circles Investigation
SA Mathematics Olympiad Problem 2016, Round 1, Question 20
SA Mathematics Olympiad 2022, Round 2, Q25
An extension of the IMO 2014 Problem 4
Anele Clive Moli's Method: Constructing an equilateral triangle
A 1999 British Mathematics Olympiad Problem and its dual
Dirk Laurie Tribute Problem
Golden Quadrilaterals
Extangential Quadrilateral
Triangulated Tangential Hexagon theorem
Theorem of Gusić & Mladinić
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Pirate Treasure Hunt and a Generalization
A Quarter Circle Investigation, Explanation & Generalization
External Links
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Created by Michael de Villiers in 2013; modified in 2014; converted to WebSketchpad, 27 Feb 2025.