**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?

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-inclined*^{1} 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).

**Note**^{1}: *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**

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Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

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