A variation of Miquel's theorem and its generalization

In the learning activity Miquel's Theorem the following interesting result was explored and proved.
If three arbitrary points D, E, and F are constructed on the sides of any triangle ABC, with D on AB, E on BC, and F on CA, then:
i) The circumcircles of triangles ADF, BDE, and CEF are concurrent.
ii) The circumcenters of triangles ADF, BDE, and CEF form a triangle similar to △ABC.

A variation of Miquel
The following trivial converse variation of Miquel's theorem is immediately obvious:
From an arbitrary point O construct equi-inclined¹ lines to the sides (or their extensions) of △ABC, then ADOF, BEOD and CFOE are cyclic, and triangle GHI is similar to ABC.
(Note: ¹Equi-inclined lines are defined here as lines that respectively make equal angles with the sides of a polygon or some other lines).

Miquel variation for triangle

Historical Note
Miquel's theorem was apparently first stated without proof by the Scottish mathematician William Wallace in 1799, but proved by the French mathematician Auguste Miquel in 1838; hence the attributed name of the theorem.

Further generalization
Click on the 'Link to Miquel Converse Quadrilateral' button to navigate to a new sketch which shows the following more surprising result:
From point M construct equi-inclined lines to the sides (or their extensions) of quadrilateral ABCD, then AQMT, BRMQ, CSMR and DTMS are cyclic (and their circumcircles are concurrent in M), and GHIJ is similar to ABCD - see the static illustration below. Navigate to the dynamic sketch by clicking the 'Link to' button above.

Miquel variation for quadrilateral

Note that this result generalizes the results mentioned in [5] for cyclic quadrilaterals, and as mentioned in [1], this converse variation result generalizes further to any pentagon, hexagon, etc.

Another, different variation of Miquel
Click on the 'Link to Different Miquel variation' button to navigate to a new sketch for a dynamic illustration of the following:
A related, but different 'converse-like' variation of Miquel’s theorem is mentioned without proof for a triangle in [2, p. 135] and [3], but proved in [4] for an n-gon by showing that for n concurrent circles after n spiral similarities of the vertices, centred at the centre of the concurrency, an identity transformation is obtained.
For example: if four circles intersect pairwise and are concurrent in M, then starting with an arbitrary point A on the circumference of one of the circles and continuing to draw straight lines through the other intersections E, F, G and H of the pairwise circles as shown in the figure below, results in A, H and D being collinear, i.e. the construction produces a closed quadrilateral. Though not explicitly mentioned in [4], it is obvious as already shown with the quadrilateral result proven in the preceding result, that the closed quadrilateral ABCD is similar to the one formed by the centres of the four circles, namely, PQRS.

Another different variation of Miquel's theorem

Note that the dynamic sketch for this variation is not very robust and will 'fall apart' due to various construction restrictions of the software, e.g. when dragging point A past point E, etc.

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
1. M. de Villiers, A variation of Miquel's theorem and its generalization, Mathematical Gazette, (July 2014), 98(542), July 2014, pp. 334-339.
2. R.A Johnson, Advanced Euclidean Geometry (Modern Geometry), New York: Dover Publications, (1929), pp. 131-135.
3. D. Wells, Penguin Dictionary of Curious and Interesting Geometry, London: Penguin Books, (1991), p. 184.
4. I.M. Yaglom (translated by A. Shields), Geometric Transformations II, Washington: The Mathematical Association of America, (1968), pp. 46-47, 129-130.
5. C. Pritchard, A tour around the geometry of a cyclic quadrilateral, talk given at the University of KwaZulu-Natal (Edgewood Campus), 12 April 2013, Slides 6-10, PDF of PowerPoint.

Other examples involving equi-inclined lines
A generalization of Neuberg's Pedal Theorem & the Simson-Wallace line
A generalization of Neuberg's Pedal Theorem to polygons
Further generalizations of Viviani's Theorem
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral

Related Links
Miquel's Theorem (Rethinking Proof activity)
Minimum Area of Miquel Circles Centres Triangle
Napoleon's Theorem (Rethinking Proof activity)
Some Triangle Generalizations of Napoleon's Theorem
(Specifically see 'A special case of Generalization 2: Miquel's Theorem')
A generalization of Neuberg's Theorem & the Simson-Wallace line
A generalization of Neuberg's Theorem to polygons
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem (Equilateral or equi-angled polygons or polygons with opposite sides parallel)
Further generalizations of Viviani's Theorem (Using equi-inclined lines)
Carnot's (or Bottema's) Perpendicularity Theorem & Some Generalizations
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
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 Theorem (Wolfram MathWorld)
Spiral Similarity (Wolfram MathWorld)
Miquel's Point: What Is It? A Mathematical Droodle (Cut The Knot)
William Wallace (mathematician) (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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