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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 supplementary angles MA´C , MB´A and MC´B .
The point at infinity along the Newton-Gauss line is the isogonal conjugate of the Miquel point. Generalization Dao Thanh Oai showed a generalization of the Newton ...
Then any point P associated with the reference triangle ABC can be defined in a Cartesian system as a vector = +. If this point P has trilinear coordinates x : y : z then the conversion formula from the coefficients k 1 and k 2 in the Cartesian representation to the trilinear coordinates is, for side lengths a, b, c opposite vertices A, B, C ,
Spiral similarity can be used to prove Miquel's Quadrilateral Theorem: given four noncollinear points ,,, and , the circumcircles of the four triangles , , , and intersect at one point, where is the intersection of and and is the intersection of and (see diagram).
Miquel configuration Rhombic dodecahedral graph. In geometry, the Miquel configuration is a configuration of eight points and six circles in the Euclidean plane, with four points per circle and three circles through each point. [1] Its Levi graph is the Rhombic dodecahedral graph, the skeleton of both Rhombic dodecahedron and Bilinski dodecahedron.
A complete quadrangle (at left) and a complete quadrilateral (at right).. In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points.
Theorem (Miquel): For the Minkowski plane () the following is true: If for any 8 pairwise not parallel points P 1 , . . . , P 8 {\displaystyle P_{1},...,P_{8}} which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples, then the sixth quadruple of points is concyclical, too.
Common nine-point circle, where N, O 4, A 4 are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points A 1, A 2, A 3. The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the ...