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From point *M* construct *equi-inclined* lines to the sides (or their extensions) of △*ABC*, then *ADMF*, *BEMD* and *CFME* are cyclic and their circumcircles are concurrent in *M*, and triangle *GHI* is similar to *ABC*.

Miquel variation for triangle

**Further generalization**: 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*.

Miquel variation for quadrilateral

The result generalizes further to any pentagon, hexagon, etc. Read my paper A variation of Miquel's theorem and its generalization in the July 2014 issue of the *Mathematical Gazette*.

**Another, different variation of Miquel**

A related, but different 'converse-like' variation of Miquel’s theorem is mentioned without proof for a triangle in [1, p. 135] and [2], but proved in [3] 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 [3], 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*.

*References*

1. R.A Johnson, *Advanced Euclidean Geometry (Modern Geometry)*, New York: Dover Publications, (1929), pp. 131-135.

2. D. Wells, *Penguin Dictionary of Curious and Interesting Geometry*, London: Penguin Books, (1991), p. 184.

3. I.M. Yaglom (translated by A. Shields), *Geometric Transformations II*, Washington: The Mathematical Association of America, (1968), pp. 46-47, 129-130.

Another different variation of Miquel's theorem

Note that in the dynamic sketch above points *A*, *B*, *C* and *D* also determine the size of the circles; so the circles will change with dragging. However, a similar sketch can easily be created with *A* chosen so that one can drag it along a stationary, fixed circle.

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By Michael de Villiers. Created, 31 October 2013; modified 23 May 2015.