The
diameter of the circumcircle, called the
circumdiameter and equal to twice the
circumradius, can be computed as the length of any side of the triangle divided by the
sine of the opposite
angle: :\text{diameter} = \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. As a consequence of the
law of sines, it does not matter which side and opposite angle are taken: the result will be the same. The diameter of the circumcircle can also be expressed as :\begin{align} \text{diameter} & {}= \frac{abc}{2\cdot\text{area}} = \frac{2|\Delta ABC|} \\[5pt] & {}= \frac{abc}{2\sqrt{s(s - a)(s - b)(s - c)}}\\[5pt] & {}= \frac{2abc}{\sqrt{(a + b + c)(-a + b + c)(a - b + c)(a + b - c)}} \end{align} where are the lengths of the sides of the triangle and s=\tfrac{a+b+c}{2} is the
semiperimeter. The expression \scriptstyle \sqrt{s(s-a)(s-b)(s-c)} above is the area of the triangle, by
Heron's formula. Trigonometric expressions for the diameter of the circumcircle include :\text{diameter} = \sqrt{\frac{2 \cdot \text{area}}{\sin A \sin B \sin C}}. The triangle's
nine-point circle has half the diameter of the circumcircle. In any given triangle, the circumcenter is always collinear with the
centroid and
orthocenter. The line that passes through all of them is known as the
Euler line. The
isogonal conjugate of the circumcenter is the
orthocenter. The useful
minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle). It is common to confuse the minimum bounding circle with the circumcircle. The circumcircle of three
collinear points is the line on which the three points lie, often referred to as a
circle of infinite radius. Nearly collinear points often lead to
numerical instability in computation of the circumcircle. Circumcircles of triangles have an intimate relationship with the
Delaunay triangulation of a
set of points. By
Euler's theorem in geometry, the distance between the circumcenter and the
incenter is :\overline{OI} = \sqrt{R(R - 2r)}, where is the incircle radius and is the circumcircle radius; hence the circumradius is at least twice the inradius (
Euler's triangle inequality), with equality only in the
equilateral case. The distance between and the
orthocenter is :\overline{OH} = \sqrt{R^2 - 8R^2\cos A \cos B \cos C} = \sqrt{9R^2 - (a^2 + b^2 + c^2)}. For
centroid and
nine-point center we have :\begin{align} \overline{IG} & The product of the incircle radius and the circumcircle radius of a triangle with sides is :rR = \frac{abc}{2(a + b + c)}. With circumradius , sides , and
medians , we have :\begin{align} 3\sqrt{3}R &\geq a + b + c \\[5pt] 9R^2 &\geq a^2 + b^2 + c^2 \\[5pt] \frac{27}{4}R^2 &\geq m_a^2 + m_b^2 + m_c^2. \end{align} If median , altitude , and internal bisector all emanate from the same vertex of a triangle with circumradius , then :4R^2 h^2(t^2 - h^2) = t^4(m^2 - h^2).
Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the
inradius. Here a segment's length is considered to be negative
if and only if the segment lies entirely outside the triangle. If a triangle has two particular circles as its circumcircle and
incircle, there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle as a vertex. (This is the case of
Poncelet's porism). A necessary and sufficient condition for such triangles to exist is the above equality \overline{OI}=\sqrt{R(R-2r)}. == Cyclic polygons ==