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.
(Below is a dynamic geometry sketch illustrating the theorem - also see A variation of Miquel's theorem, which shows a variation/converse).

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 (area ∆ABC)/(area ∆GHI) in that case?

.sketch_canvas { border: medium solid lightgray; display: inline-block; } 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 & instead start with an arbitrary point M and draw equi-inclined1 lines towards the sides of a quadrilateral ABCD, then the centres of the formed circumcircles form a quadrilateral GHIJ similar to ABCD (De Villiers, 2014).
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 (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?

Reference
De Villiers, M. (2014). A variation of Miquel’s theorem and its generalization. The Mathematical Gazette, 98(542), 334-339.

Submitted Paper
A paper "Optimizing Triangle Areas: Miquel Circles Center Triangle and 3rd Pedal Triangle" by Hans Humenberger & myself, with proofs of these results, has been submitted for publication. All Rights Reserved.

Related Links
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
Equi-inclined Lines Problem
Generalizations of a theorem by Wares

Created 23 July 2023 by Michael de Villiers, using WebSketchpad.