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[[File:Miquel Circles.svg|frame|A diagram showing circles passing through the vertices of a triangle ''ABC'' and points ''A´'', ''B´'' and ''C´'' on the adjacent sides of the triangle intersecting at a common point, ''M''.]]
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'''Miquel's theorem''' is a result in [[geometry]], named after [[Auguste Miquel]], concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides.  
 
Formally, let ''ABC'' be a triangle, with points ''A´'', ''B´'' and ''C´'' on sides ''BC'', ''AC'', and ''AB'' respectively. Draw three [[circumcircle]]s to triangles ''AB´C´'', ''A´BC´'', and ''A´B´C''. Miquel's theorem then states that these circles intersect in a single point ''M'', the '''Miquel point'''. In addition, the three angles ''MA´B'', ''MB´C'' and ''MC´A'' (green in the diagram) are all equal, as are the three complementary angles ''MA´C'', ''MB´A'' and ''MC´B''.
 
The theorem (and its corollary) follow from the properties of two [[cyclic quadrilateral]]s drawn from any two of a triangle's vertices, having an edge in common as shown in the figure. Their combined angles at ''M'' (opposite ''A'' and opposite ''C'') will be (180 - ''A'') + (180 - ''C''), giving an exterior angle equal to (''A'' + ''C''). Since (''A'' + ''C'') also equals (180 - ''B''), the intersection at ''M'', lying on the chord ''A´C´'', must also lie on a cyclic quadrilateral passing through points ''B'', ''A´'', and ''C´''. This completes the proof.
 
If the fractional distances of ''A´'', ''B´'' and ''C´'' along sides ''BC'' (''a''), ''CA'' (''b'') and ''AB'' (''c'') are ''d''<sub>''a''</sub>, ''d''<sub>''b''</sub> and ''d''<sub>''c''</sub>,  respectively, the Miquel point, in [[trilinear coordinates]] (''x'' : ''y'' : ''z''), is given by:
 
:<math>x=a \left(-a^2 d_a d_a^{\,'} + b^2 d_a d_b + c^2 d_a^{\,'} d_c^{\,'} \right)</math>
:<math>y=b \left(a^2 d_a^{\,'} d_b^{\,'} - b^2 d_b d_b^{\,'} + c^2 d_b d_c \right)</math>
:<math>z=c \left(a^2 d_a d_c + b^2 d_b^{\,'} d_c^{\,'} - c^2 d_c d_c^{\,'} \right),</math>
 
where ''d'''<sub>''a''</sub> = 1 - ''d''<sub>''a''</sub>, ''etc.''
 
In the case ''d''<sub>''a''</sub> = ''d''<sub>''b''</sub> = ''d''<sub>''c''</sub> = ½ the Miquel point is the [[circumcentre]] {{math|(cos &alpha; : cos &beta; : cos &gamma;)}}.
 
The theorem can be reversed to say: for three circles intersecting at ''M'', a line can be drawn from any point ''A'' on one circle, through its intersection ''C´'' with another to give ''B'' (at the second intersection). ''B'' is then similarly connected, via intersection at ''A´'' of the second and third circles, giving point ''C''. Points ''C'', ''A'' and the remaining point of intersection, ''B´'',  will then be collinear, and triangle ''ABC'' will always pass though the circle intersections ''A´'', ''B´'' and ''C´''.
 
[[File:Miquel's Theorem 2.svg|frame|'''Miquel's six circles theorem''' states that if five circles share four triple-points of intersection then the remaining four points of intersection lie on a sixth circle.]]
This can be extended to a circle with four points. Given points, ''A'', ''B'', ''C'', and ''D'' on a circle, and circles passing through each adjacent pair of points, the alternate intersections of these four circles at ''W'', ''X'', ''Y'' and ''Z'' then lie on a common circle. This is known as '''Miquel's six circles theorem'''.
 
==See also==
* [[Pivot theorem]]
* [[Clifford's circle theorems]]
* [[Five circles theorem]]
 
==Bibliography==
* {{citation | last = Wells | first = David | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 151–152}}
* {{citation|first=Auguste|last= Miquel|title=Mémoire de Géométrie|journal=[[Journal de Mathématiques Pures et Appliquées]]|volume= 1|pages= 485–487|year= 1838|url=http://mathdoc.emath.fr/JMPA/feuilleter.php?id=JMPA_1838_1_3}}.
 
==External links==
* {{MathWorld|title=Miquel's theorem|urlname=MiquelsTheorem}}
* [http://math.kennesaw.edu/~mdevilli/napole-general.html Miquels' Theorem as a special case of a generalization of Napoleon's Theorem] at [http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm Dynamic Geometry Sketches]
 
[[Category:Circles]]
[[Category:Theorems in geometry]]

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