**NOTE**: Please WAIT while the Applet below loads.

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

**Note**: *Equi-inclined* lines are lines that respectively make *equal angles* with some other lines.

Miquel variation for triangle

**Important**: To view & manipulate the *dynamic versions* of the two generalizations below, navigate to them using the appropriate Link button in the **ABOVE** dynamic sketch; the pictures below are **static**.

**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 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.

Other examples of results involving *equi-inclined* lines:

A generalization of Neuberg's Theorem and the Simson line

Equi-inclined Lines Problem

Generalizations of a theorem by Wares

Copyright © 2016 KCP Technologies, a McGraw-Hill Education Company. All rights reserved.

Release: 2017Q2beta, Semantic Version: 4.5.1-alpha, Build Number: 1021, Build Stamp: ip-10-149-70-76/20160706123640

*Back
to "Dynamic Geometry Sketches"*

*Back
to "Student Explorations"*

Created by Michael de Villiers, 31 October 2013. Modified 23 May 2015; 17 April 2020.