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- Miquels_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. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal Journal de mathématiques pures et appliquées.Formally, let ABC be a triangle, with arbitrary points A´, B´ and C´ on sides BC, AC, and AB respectively (or their extensions). Draw three circumcircles to triangles AB´C´, A´BC´, and A´B´C. Miquel's theorem states that these circles intersect in a single point M, called 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.".
- Miquels_theorem thumbnail Miquel_Circles.svg?width=300.
- Miquels_theorem wikiPageExternalLink JavaGSPLinks.htm.
- Miquels_theorem wikiPageExternalLink napole-general.html.
- Miquels_theorem wikiPageID "19335221".
- Miquels_theorem wikiPageLength "8637".
- Miquels_theorem wikiPageOutDegree "30".
- Miquels_theorem wikiPageRevisionID "705847720".
- Miquels_theorem wikiPageWikiLink Auguste_Miquel.
- Miquels_theorem wikiPageWikiLink Bundle_theorem.
- Miquels_theorem wikiPageWikiLink Category:Circles.
- Miquels_theorem wikiPageWikiLink Category:Theorems_in_geometry.
- Miquels_theorem wikiPageWikiLink Category:Theorems_in_plane_geometry.
- Miquels_theorem wikiPageWikiLink Circumscribed_circle.
- Miquels_theorem wikiPageWikiLink Cliffords_circle_theorems.
- Miquels_theorem wikiPageWikiLink Collinearity.
- Miquels_theorem wikiPageWikiLink Complete_quadrangle.
- Miquels_theorem wikiPageWikiLink Cyclic_quadrilateral.
- Miquels_theorem wikiPageWikiLink Euclidean_geometry.
- Miquels_theorem wikiPageWikiLink Five_circles_theorem.
- Miquels_theorem wikiPageWikiLink Geometry.
- Miquels_theorem wikiPageWikiLink Jakob_Steiner.
- Miquels_theorem wikiPageWikiLink Joseph_Diaz_Gergonne.
- Miquels_theorem wikiPageWikiLink Joseph_Liouville.
- Miquels_theorem wikiPageWikiLink Journal_de_Mathématiques_Pures_et_Appliquées.
- Miquels_theorem wikiPageWikiLink Mathematical_Association_of_America.
- Miquels_theorem wikiPageWikiLink School_Mathematics_Study_Group.
- Miquels_theorem wikiPageWikiLink Trilinear_coordinates.
- Miquels_theorem wikiPageWikiLink Washington,_D.C..
- Miquels_theorem wikiPageWikiLink File:Miquel5.svg.
- Miquels_theorem wikiPageWikiLink File:Miquel_Circles.svg.
- Miquels_theorem wikiPageWikiLink File:Miquelquad.svg.
- Miquels_theorem wikiPageWikiLink File:Miquels_Theorem_2.svg.
- Miquels_theorem wikiPageWikiLink File:Pivot_theorem.svg.
- Miquels_theorem wikiPageWikiLink File:Pivot_theorem_3d.png.
- Miquels_theorem wikiPageWikiLinkText "Miquel's theorem".
- Miquels_theorem wikiPageWikiLinkText "Miquel's theorem#Pivot theorem".
- Miquels_theorem wikiPageWikiLinkText "Pivot theorem".
- Miquels_theorem wikiPageWikiLinkText "here".
- Miquels_theorem title "Miquel Five Circles Theorem".
- Miquels_theorem title "Miquel Pentagram Theorem".
- Miquels_theorem title "Miquel's theorem".
- Miquels_theorem title "Pivot theorem".
- Miquels_theorem urlname "MiquelFiveCirclesTheorem".
- Miquels_theorem urlname "MiquelsPentagramTheorem".
- Miquels_theorem urlname "MiquelsTheorem".
- Miquels_theorem urlname "PivotTheorem".
- Miquels_theorem wikiPageUsesTemplate Template:Citation.
- Miquels_theorem wikiPageUsesTemplate Template:Commons_category.
- Miquels_theorem wikiPageUsesTemplate Template:Harvtxt.
- Miquels_theorem wikiPageUsesTemplate Template:Math.
- Miquels_theorem wikiPageUsesTemplate Template:MathWorld.
- Miquels_theorem wikiPageUsesTemplate Template:Reflist.
- Miquels_theorem subject Category:Circles.
- Miquels_theorem subject Category:Theorems_in_geometry.
- Miquels_theorem subject Category:Theorems_in_plane_geometry.
- Miquels_theorem hypernym Result.
- Miquels_theorem type Redirect.
- Miquels_theorem type Theorem.
- Miquels_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.".
- Miquels_theorem label "Miquel's theorem".
- Miquels_theorem sameAs Q2449984.
- Miquels_theorem sameAs Satz_von_Miquel.
- Miquels_theorem sameAs Théorème_de_Miquel.
- Miquels_theorem sameAs Pivoteerpuntstelling_van_Miquel.
- Miquels_theorem sameAs m.04n561n.
- Miquels_theorem sameAs Точка_Микеля.
- Miquels_theorem sameAs Định_lý_Miquel.
- Miquels_theorem sameAs Q2449984.
- Miquels_theorem sameAs 密克定理.
- Miquels_theorem wasDerivedFrom Miquels_theorem?oldid=705847720.
- Miquels_theorem depiction Miquel_Circles.svg.
- Miquels_theorem isPrimaryTopicOf Miquels_theorem.