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...

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...

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....

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...

where R {\displaystyle R} and r {\displaystyle r} are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the...

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...

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...

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...

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...

same side of length f and share the same circumcircle, so that the side of length f is a chord of the circumcircle and the triangles have side lengths (a...

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...

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...

Archimedes' twin circles Bankoff circle Circular triangle Reuleaux triangle Circumcircle Disc Incircle and excircles of a triangle Nine-point circle Circular...

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...

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...

the radius in terms of the coordinates of the three given points. See circumcircle. Chords are equidistant from the centre of a circle if and only if they...

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...

for example, AD, is extended to intersect the circumcircle at P, so that AD is a chord of the circumcircle, then the foot D bisects segment HP: H D ¯ =...

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

{MI}}=2{\sqrt {R^{2}-{\frac {abc}{a+b+c}}}}} where R denotes the radius of the circumcircle and a, b, c the sides of △ABC. The Bevan is point is also the midpoint...

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...

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