In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called...

right angle at the apex and the hypotenuse as the base; conversely, the circumcircle of any right triangle has the hypotenuse as its diameter. This is Thales'...

length of the sides equals the radius of the circumscribed circle or circumcircle, which equals 2 3 {\displaystyle {\tfrac {2}{\sqrt {3}}}} times the apothem...

their convex hull into triangles whose circumcircles do not contain any of the points; that is, each circumcircle has its generating points on its circumference...

circle, making the sides chords of the circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The...

circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing...

given circumcircle may be constructed as follows: Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle. Draw...

cyclic polygon, and the circle is called its circumscribing circle or circumcircle. All concyclic points are equidistant from the center of the circle....

circle) also passing through two triangle vertices with its center on the circumcircle. This theorem is best known in Russia, where it is called the trillium...

point, the triangle's circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. Thales' theorem implies...

opposite angles (see figure 2), while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes...

new points to form the decagon. Both in the construction with given circumcircle as well as with given side length is the golden ratio dividing a line...

perpendicular bisectors of its three sides, which is also the center of the circumcircle that passes through the three vertices). In an isosceles triangle with...

circle touch is called the Feuerbach point. The radius of a triangle's circumcircle is twice the radius of that triangle's nine-point circle.: p.153  Figure...

formed from them; such a polygon is said to be inscribed in the circle. Circumcircle, the circumscribed circle of a triangle, which always exists for a given...

In geometry, given a triangle ABC and a point P on its circumcircle, the three closest points to P on lines AB, AC, and BC are collinear. The line through...

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...

quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius...

triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the tangential triangle coincides with the circumcircle of the reference...

other) that can be inscribed in a circle. That is, it is a kite with a circumcircle (i.e., a cyclic kite). Thus the right kite is a convex quadrilateral...

perpendiculars from a regular n-gon's vertices to any line tangent to the circumcircle equals n times the circumradius.: p. 73  The sum of the squared distances...

in the work of Albrecht Dürer. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then B D = 1 2 B C {\displaystyle \scriptstyle {BD={1...

subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a star-shaped polygonal hole...

C meet the circumcircle of △ABC at A', B', C' respectively. Let △DEF be the triangle formed by the tangents at A, B, C to the circumcircle of △ABC. (Let...

circle is called the circumcircle of the triangle. One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle...

cyclic polygon, meaning that its vertices all lie on a common circle. The circumcircle of the Lemoine hexagon is known as the first Lemoine circle. Casey, John...

nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius. Denote the circumradius of the triangle...

in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length. Both of the above displayed different...

according as the line intersects the circumcircle in 0, 1, or 2 points. The isogonal conjugate of the circumcircle is the line at infinity. Several well-known...

thing, by analogy with the term circumcircle. As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed...