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...

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

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...

whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The...

points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points. This maximizes the size of the smallest...

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...

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...

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...

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...

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...

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...

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

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...

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...

is the circumcenter, then △LMN is the medial triangle. If P is on the circumcircle of the triangle, △LMN collapses to a line (the pedal line or Simson line)...

pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE. For an arbitrary...

diagonal, and is equal to 2 . {\displaystyle {\sqrt {2}}.} Then the circumcircle has the equation x 2 + y 2 = 2. {\displaystyle x^{2}+y^{2}=2.} Alternatively...

of an equilateral triangle A B C {\displaystyle ABC} but not on its circumcircle, then there exists a triangle with sides of lengths P A {\displaystyle...

As a corollary of the chord formula, the area bounded by the circumcircle and incircle of every unit convex regular polygon is π/4...

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...

triangle, two concentric circles (with that distance being zero) are the circumcircle and incircle of a triangle if and only if the radius of one is twice...

sides equal. A rhombus has an inscribed circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex...

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

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...

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...

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...

are equal. Its centre is equidistant from its vertices, hence it has a circumcircle. Its centre is equidistant from its sides, hence it has an incircle....

An animation of a neusis construction of a heptagon with radius of circumcircle O A ¯ = 6 {\displaystyle {\overline {OA}}=6} , based on Andrew M. Gleason...