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- Miquel's_theorem abstract "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 circumcircles 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 quadrilaterals 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 da, db and dc, respectively, the Miquel point, in trilinear coordinates (x : y : z), is given by:where d'a = 1 - da, etc.In the case da = db = dc = ½ the Miquel point is the circumcentre (cos α : cos β : cos γ).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´.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.".
- Miquel's_theorem thumbnail Miquel_Circles.svg?width=300.
- Miquel's_theorem wikiPageExternalLink JavaGSPLinks.htm.
- Miquel's_theorem wikiPageExternalLink napole-general.html.
- Miquel's_theorem wikiPageExternalLink feuilleter.php?id=JMPA_1838_1_3.
- Miquel's_theorem wikiPageID "19335221".
- Miquel's_theorem wikiPageRevisionID "593024398".
- Miquel's_theorem hasPhotoCollection Miquel's_theorem.
- Miquel's_theorem title "Miquel's theorem".
- Miquel's_theorem urlname "MiquelsTheorem".
- Miquel's_theorem subject Category:Circles.
- Miquel's_theorem subject Category:Theorems_in_geometry.
- Miquel's_theorem type Abstraction100002137.
- Miquel's_theorem type Attribute100024264.
- Miquel's_theorem type Circle113873502.
- Miquel's_theorem type Circles.
- Miquel's_theorem type Communication100033020.
- Miquel's_theorem type ConicSection113872975.
- Miquel's_theorem type Ellipse113878306.
- Miquel's_theorem type Figure113862780.
- Miquel's_theorem type Message106598915.
- Miquel's_theorem type PlaneFigure113863186.
- Miquel's_theorem type Proposition106750804.
- Miquel's_theorem type Shape100027807.
- Miquel's_theorem type Statement106722453.
- Miquel's_theorem type Theorem106752293.
- Miquel's_theorem type TheoremsInGeometry.
- Miquel's_theorem comment "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 circumcircles 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.".
- Miquel's_theorem label "Miquel's theorem".
- Miquel's_theorem label "Pivoteerpuntstelling van Miquel".
- Miquel's_theorem label "Satz von Miquel".
- Miquel's_theorem label "Théorème de Miquel".
- Miquel's_theorem label "Точка Микеля".
- Miquel's_theorem label "密克定理".
- Miquel's_theorem sameAs Satz_von_Miquel.
- Miquel's_theorem sameAs Théorème_de_Miquel.
- Miquel's_theorem sameAs Pivoteerpuntstelling_van_Miquel.
- Miquel's_theorem sameAs m.04n561n.
- Miquel's_theorem sameAs Q2449984.
- Miquel's_theorem sameAs Q2449984.
- Miquel's_theorem sameAs Miquel's_theorem.
- Miquel's_theorem wasDerivedFrom Miquel's_theorem?oldid=593024398.
- Miquel's_theorem depiction Miquel_Circles.svg.
- Miquel's_theorem isPrimaryTopicOf Miquel's_theorem.
