Miquel's Theorem

The dynamic geometry activities below are from the "Proof as Challenge" 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: Miquel Worksheet & Teacher Notes.

Prerequisites
Knowledge of the properties of cyclic quadrilaterals, the AA condition of similarity, and the fact that the diagonals of a kite are perpendicular to each other. Specifically, it is highly recommended that students have already completed the following learning activities:
Exploring Kite Properties (Suggested Grades 4 - 9)
Cyclic Quadrilateral (Opposite ∠'s supplementary)
Cyclic Quadrilateral Converse (Converse of result above)

Miquel's Theorem
In this investigation, you will explore a construction based on arbitrary points on the sides of an arbitrary triangle and some circles related to these points. The result you will find was apparently first discovered by a French mathematician named Auguste Miquel in 1838.

Miquel's Theorem

Note
1) Users are strongly encouraged to download the accompanying Worksheet, and to print it out to use in conjunction with the dynamic sketch above. Alternatively, copy the URL: https://dynamicmathematicslearning.com/miquel-rethinking-proof.pdf and paste it into a new browser window. Then resize the new window to place it side by side with this one, or one below the other. (However, this is not likely to be a feasible option for users using small screens such as a cellphone or tablet.)

Explore
2) Drag any of points A, B, C, D, E or F to familiarize yourself with the sketch.
3) Press the 'Show circle FEC & center I' button to show circumcircle I of △FEC.
4) What do you notice about the three circumcircles? Check your observation by dragging.
5) Does your observation hold if any or all of the arbitrary points D, E or F are dragged on to the extensions of the sides?
Conjecture 1
6) Formulate a conjecture based on your observations in 3)-5).
7) Press the 'Show △GHI' button to show △GHI.
8a) What do you notice about the shape of △GHI in relation to △ABC? Check your intuitive guess by dragging any of the red vertices.
Measure the size of the angles of the two triangles by the 'Angle' Tool in the Toolbar on the left to check your intuitive guess above.
8b) Alternatively to 8a), measure the side lengths of the two triangles by using the 'Distance' Tool in the Toolbar on the left. Then calculate the ratios between corresponding sides by using the 'Calculate' Tool in the Toolbar on the left (scroll down if necessary).
Conjecture 2
9) What do you notice in 8) above? Can you formulate a conjecture? Does your conjecture still hold if any or all of the arbitrary points D, E or F are dragged on to the extensions of the sides?
10) Check your conjectures in 6), 8) and 9) by clicking on the 'Link to Miquel Conjectures' button to navigate to a new sketch.

Challenge
11) Can you prove your two conjectures in 10) above? Can you prove them in more than one way?

Proving
12) Work through the questions in the guided proofs in the accompanying Worksheet, using the four top buttons on the right as indicated in the text (the 'Show segments' button as well as the three 'Show quadrilateral' buttons). The proofs require only knowledge of the properties of cyclic quadrilaterals and kites.

Investigate Further
What happens if instead of starting with arbitrary points D, E and F on the sides, we instead start with an arbitrary point O in a triangle ABC and construct lines from O to make equal angles with the sides?
13) To investigate this situation, click on the 'Link to Miquel Converse' button to navigate to a new sketch.
14) What do you notice? Can you prove your observations?
Explore More
What happens if we start with an arbitrary point O in a quadrilateral ABCD and construct equi-inclined lines from O to the four sides of ABCD? (Equi-inclined lines make equal angles with the sides).
15) To explore this question go to: A variation of Miquel's theorem and its generalization.

Connection to Napoleon's theorem
Miquel's theorem is very closely connected to Napoleon's Theorem. In fact, one can view Miquel's theorem as a special case of a generalization of Napoleon's Theorem as shown at either one of the following dynamic webpages:
Some Triangle Generalizations of Napoleon's Theorem (specifically look at 'A special case of Generalization 2: Miquel's Theorem')
Some Circle Concurrency Theorems (specifically look at 'Miquel's Theorem')

Explore minimum area of ΔGHI
What is the minimum area of ΔGHI in relation to a fixed ΔABC? Dynamically explore this interesting question at the link below.
Minimum Area of Miquel Circles Centres Triangle

References
Coxeter, H. & Greitzer, S. (1967). Geometry Revisited, pp. 61-65. Washington: DC, MAA.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2014). A variation of Miquel's theorem and its generalization. The Mathematical Gazette, 98(542), 334-339.
De Villiers, M. (2022). Some Circle Concurrency Theorems. Learning and Teaching Mathematics, No. 33 Dec), pp. 34-38.
Humenberger, H. & De Villiers, M. (2026). Optimising Miquel circles centre triangles and third pedal triangles. The Mathematical Gazette. DOI: 10.1080/00255572.2025.2539599
Rong, V. (2021). Complete Quadrilaterals and the Miquel Point. Summer (Camp) 2021.
Weiss, G. & Odehnal, B. (2024). Miquel’s Theorem and its Elementary Geometric Relatives. KoG. DOI:10.31896/k.28.2.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Miquel's theorem (Wikipedia)
Miquel's Theorem (Wolfram MathWorld)
Miquel's Point: What Is It? A Mathematical Droodle (Cut The Knot)
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, 10 April 2026.