Matches in DBpedia 2016-04 for { ?s ?p "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."@en }
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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.".
- Q2449984 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.".