Isosceles Trapezoid Midpoints

The dynamic geometry activities below are from the "Proof as Discovery" section of my book Rethinking Proof (free to download).

Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Isosceles Trapezoid Midpoints Worksheet & Teacher Notes.

Prerequisites
Students need to know that the length of a triangle midsegment is half the length of the corresponding base. They should've already discovered this property in a preceding activity Kite Midpoints. In the activity below, this property is assumed without proof, but it is later proved in the Systematization section in the activity Reasoning Backwards: Triangle Midpoints. It is also recommended that students have already completed the following activities:
Visually Introducing & Classifying an Isosceles Trapezoid (Trapezium) (Suggested Grades 1 - 7)
Exploring Isosceles Trapezoid (Trapezium) Properties (Suggested Grades 4 - 9)
Exploring Rhombus Properties (Suggested Grades 4 - 9)

The dynamic sketch below shows an isosceles trapezoid ABCD which you will use in the guided activity below to discover, prove (and generalize) an interesting property about the quadrilateral formed by the midpoints of its triangles.

Isosceles Trapezoid Midpoints

Explore
1) Click 'Show midpoint quadrilateral' button in the sketch.
2) What kind of quadrilateral is the midpoint quadrilateral EFGH? Drag vertices to test your conjecture.
3) Press the 'Show diagonals' button to show the diagonals of ABCD. This button also displays the measure of the angle at the intersection of the diagonals. What connection or relationship do you see between the formed quadrilateral and the diagonals? Check by dragging.
4) Does your conjecture in 2) and 3) still hold if you drag ABCD into the shape of a crossed quadrilateral?
5) Drag any of the points A, B, C, and D. Can EFGH ever be a square? If so, when?

Challenge
6) Can you explain why your conjectures from Questions 2) to 5) are true? Try to construct your own logical explanations before reading ahead to the hints that follow.

(Logically) Explaining
You should have discovered the following two conjectures in your investigation:
.....a) The midpoint quadrilateral of an isosceles trapezoid is a rhombus.
.....b) The midpoint quadrilateral of an isosceles trapezoid is a square only when the diagonals of the isosceles trapezoid are perpendicular to each other.
Although you are no doubt already convinced about these observations, can you explain, in terms of other geometric results, why your observations are true? Below are some questions for planning logical explanations of your observations. You will begin by explaining the first conjecture. Then in Question 14 you will explain the second conjecture.
7) What is the relationship between EF and AC in △ACB? Why?
8) What is the relationship between HG and AC in △ACD? Why?
9) From Questions 7) and 8), what can you therefore conclude regarding EF and HG?
10) What is the relationship between EH and BD in △ABD? Why?
11) What is the relationship between FG and BD in △CBD? Why?
12) From Questions 10) and 11), what can you conclude about EH and FG?
13) Given that the lengths of the diagonals of an isosceles trapezoid are always equal (as explored in Exploring Isosceles Trapezoid Properties), what can you now conclude about the relationships between the adjacent sides of quadrilateral EFGH?
14) If AC ⊥ BD, what can you then say about the angles of EFGH? Why?

Looking back & Discovering
15) Apart from the property of congruent diagonals, did you use any other property exclusive to isosceles trapezoids for your conclusion in Question 13? (For example, did you use the property that an isosceles trapezoid has (at least) one pair of parallel sides or (at least) one line of symmetry through a pair of opposite sides?)
16) Use Question 15) to describe the most generic quadrilateral that will always have a rhombus as its midpoint quadrilateral.
17) So far we’ve seen that new results in mathematics can be discovered by experimentation, logical reasoning, or careful reflection on logical explanations. Which of these three different ways best describes your discovery in Question 15 or 16?

Construction Challenge
18) Can you construct a (dynamic) generic quadrilateral whose midpoint quadrilateral is always a rhombus? Click on the 'Link to equidiagonal quad' button to navigate to a new sketch and use the tools on the left (scroll down if necessary) to construct such a generic quadrilateral with its midpoint quadrilateral always a rhombus.
19) Can you construct a (dynamic) generic quadrilateral whose midpoint quadrilateral is always a square? Can you construct a generic quadrilateral whose midpoint quadrilateral is always a rhombus? Click on the 'Link to midsquare quad' button to navigate to a new sketch and use the tools on the left (scroll down if necessary) to construct such a generic quadrilateral with its midpoint quadrilateral always a rhombus.

Construction Solutions
Check or compare your construction solutions to Questions 18) and 19) with those given here: Construction Solutions.

Reference
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Lulu Press: Dynamic Mathematics Learning.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Isosceles trapezoid (Wikipedia)
Equidiagonal quadrilateral (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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 with WebSketchpad, 2 May 2026.