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  2. Circumcircle - Wikipedia

    en.wikipedia.org/wiki/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.

  3. Thales's theorem - Wikipedia

    en.wikipedia.org/wiki/Thales's_theorem

    This point must be equidistant from the vertices of the triangle.) This 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 then the triangle is right, and the center of its circumcircle lies on its hypotenuse.

  4. Circumscribed circle - Wikipedia

    en.wikipedia.org/wiki/Circumscribed_circle

    Circumcircle, the circumscribed circle of a triangle, which always exists for a given triangle. Cyclic polygon, a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic points. All triangles are cyclic polygons. Cyclic quadrilateral, a special case of a cyclic polygon.

  5. Incircle and excircles - Wikipedia

    en.wikipedia.org/wiki/Incircle_and_excircles

    The product of the incircle radius and the circumcircle radius of a triangle with sides , , and is [13] = (+ +). Some relations among the sides, incircle radius, and circumcircle radius are: [ 14 ] a b + b c + c a = s 2 + ( 4 R + r ) r , a 2 + b 2 + c 2 = 2 s 2 − 2 ( 4 R + r ) r . {\displaystyle {\begin{aligned}ab+bc+ca&=s^{2}+(4R+r)r,\\a^{2 ...

  6. Modern triangle geometry - Wikipedia

    en.wikipedia.org/wiki/Modern_triangle_geometry

    Two triangles are said to be poristic triangles if they have the same incircle and circumcircle. Given a circle with Center O and radius R and another circle with center I and radius r, there are an infinite number of triangles ABC with Circle O(R) as circumcircle and I(r) as incircle if and only if OI 2 = R 2 − 2Rr. These triangles form a ...

  7. Right triangle - Wikipedia

    en.wikipedia.org/wiki/Right_triangle

    The converse states that the hypotenuse of a right triangle is the diameter of its circumcircle. As a corollary, the circumcircle has its center at the midpoint of the diameter, so the median through the right-angled vertex is a radius, and the circumradius is half the length of the hypotenuse.

  8. Concyclic points - Wikipedia

    en.wikipedia.org/wiki/Concyclic_points

    Three points in the plane that do not all fall on a straight line are concyclic, so every triangle is a cyclic polygon, with a well-defined circumcircle. However, four or more points in the plane are not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied.

  9. Nine-point circle - Wikipedia

    en.wikipedia.org/wiki/Nine-point_circle

    The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's medial triangle (with vertices at the midpoints of the sides of the reference triangle) and its orthic triangle (with vertices at the feet of the reference triangle's altitudes). [6]: p.153

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