## A variation of Miquel's theorem and its generalization

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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.

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.

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.

#### Please install Java (version 1.4 or later) to use JavaSketchpad applets.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.