Minimum Area of Miquel Circle Centres Triangle

Miquel's Theorem: If arbitrary points D, E and F are respectively constructed on the lines AB, BC and CA of ∆ABC, then the circumcircles of triangles ADF, BDE and CEF are concurrent in O, and ∆GHI formed by the centres of these three circles, is similar to ∆ABC.
(For a dynamic investigation and guided proof go to: Miquel's Theorem (Rethinking Proof activity) or see De Villiers (1999, 2012). The dynamic geometry sketch below also illustrates the theorem. In addition, also see A variation of Miquel's theorem, which shows a variation/converse and further generalization).

Investigate
1) Use the sketch below and drag D, E and F to investigate where ∆GHI has minimum area. What do you notice?
2) When you have reached the minimum in 1), click on the 'Show Area Ratio' button. What do you notice?
3) Drag any of A, B, or C. Then repeat steps 1) and 2).
4) Formulate a conjecture about the location of ∆GHI to have minimum area (with respect to a fixed ∆ABC). And what is the maximum value of the ratio $$\frac{area∆ABC}{area∆GHI}$$ in that case?

Minimum Area of Miquel Circle Centres Triangle

Challenge 1
Can you explain why (prove that) that your conjecture in 4) above is true?

Further Generalization
Miquel's theorem as stated above does not generalize to polygons, i.e. if for example arbitrary points on the sides of a quadrilateral are chosen & the same circumcircle constructions as before are repeated, then the four circumcircles would not necessarily be concurrent. However, if we switch it around starting with four concurrent circles, or alternatively, instead start with an arbitrary point M and draw equi-inclined¹ lines towards the sides of a quadrilateral ABCD, then the centres of the four circumcircles form a quadrilateral GHIJ similar to ABCD (De Villiers, 2014; Yaglom, 1968).
Note¹: Equi-inclined lines are lines that respectively make equal angles with the sides of a polygon or some other lines.

5) Click on the 'Link to Miquel converse quadrilateral' button on the bottom right to view this construction for a quadrilateral.

Investigate
6) Use the sketch above and drag M or Q to investigate where quadrilateral GHIF has minimum area. What do you notice?
7) When you have reached the minimum in 1), click on the 'Show Area Ratio' button. What do you notice?
8) Drag any of A, B, C, or D. Then repeat steps 6) and 7).
9) Formulate a conjecture about the location of ∆GHIF to have minimum area (with respect to a fixed ∆ABCD). And what is the maximum value of the ratio $$\frac{area∆ABCD}{area∆GHIF}$$ in that case?

Challenge 2
Can you explain why (prove that) that your conjecture in 9) above is true? Does your argument/proof generalize to any polygon?

References
Yaglom, I. M. (1968). Geometric Transformations II. (Translated from Russian by Allen Shields). NY: Random House, Inc., pp. 129-130.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press, pp. 122-125; pp. 197-198.
De Villiers, M. (2014). A variation of Miquel's theorem and its generalization. The Mathematical Gazette, 98(542), 334-339.

Published Paper
A paper Optimising Miquel circles centre triangles and third pedal triangles by Hans Humenberger (University of Vienna) & myself, with proofs of these results, has been published Online Open Access, 19 Jan 2026, in The Mathematical Gazette. DOI: 10.1080/00255572.2025.2539599.

Related Links
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel and its generalization
A generalization of Neuberg's Theorem and the Simson line
Maximising the Area of the 3rd Pedal Triangle in Neuberg's theorem
Varignon Parallelogram Area (Rethinking Proof activity)
Area ratios of some parallel-polygons inscribed in quadrilaterals and triangles
(above webpage explores the areas of the general Varignon parallelogram & Thomsen's hexagon)
Maximizing the Area of Van Aubel's Quadrilateral
Maximizing the Area of Napoleon's Triangle
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral

External Links
Miquel's theorem (Wikipedia)
Spiral similarity (Wikipedia)
Miquel's Point (Cut The Knot)
Miquel's Theorem (Wolfram MathWorld)
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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