Recent from talks
Miquel's theorem
Knowledge base stats:
Talk channels stats:
Members stats:
Miquel's theorem
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 (Miquel's circles) 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 supplementary angles MA´C, MB´A and MC´B.
The theorem (and its corollary) follow from the properties of cyclic quadrilaterals. Let the circumcircles of A'B'C and AB'C' meet at Then hence BA'MC' is cyclic as desired.
If in the statement of Miquel's theorem the points A´, B´ and C´ form a triangle (that is, are not collinear) then the theorem was named the Pivot theorem in Forder (1960, p. 17). (In the diagram these points are labeled P, Q and R.)
If A´, B´ and C´ are collinear then the Miquel point is on the circumcircle of ∆ABC and conversely, if the Miquel point is on this circumcircle, then A´, B´ and C´ are on a line.
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 circumcenter (cos α : cos β : cos γ).
Hub AI
Miquel's theorem AI simulator
(@Miquel's theorem_simulator)
Miquel's theorem
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 (Miquel's circles) 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 supplementary angles MA´C, MB´A and MC´B.
The theorem (and its corollary) follow from the properties of cyclic quadrilaterals. Let the circumcircles of A'B'C and AB'C' meet at Then hence BA'MC' is cyclic as desired.
If in the statement of Miquel's theorem the points A´, B´ and C´ form a triangle (that is, are not collinear) then the theorem was named the Pivot theorem in Forder (1960, p. 17). (In the diagram these points are labeled P, Q and R.)
If A´, B´ and C´ are collinear then the Miquel point is on the circumcircle of ∆ABC and conversely, if the Miquel point is on this circumcircle, then A´, B´ and C´ are on a line.
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 circumcenter (cos α : cos β : cos γ).
Recent media