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  2. 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 22 ( 4 R + r ) r . {\displaystyle {\begin{aligned}ab+bc+ca&=s^{2}+(4R+r)r,\\a^{2 ...

  3. Curve of constant width - Wikipedia

    en.wikipedia.org/wiki/Curve_of_constant_width

    A standard example is the Reuleaux triangle, the intersection of three circles, each centered where the other two circles cross. [2] Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not smooth , and in fact these angles are the sharpest possible for any curve of constant width.

  4. Equation - Wikipedia

    en.wikipedia.org/wiki/Equation

    For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates x and y satisfy the equation x 2 + y 2 = 4. Parametric equations

  5. Cyclic quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Cyclic_quadrilateral

    Examples of cyclic quadrilaterals. In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral 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.

  6. Circumcircle - Wikipedia

    en.wikipedia.org/wiki/Circumcircle

    Hence, given the radius, r, center, P c, a point on the circle, P 0 and a unit normal of the plane containing the circle, ⁠ ^, ⁠ one parametric equation of the circle starting from the point P 0 and proceeding in a positively oriented (i.e., right-handed) sense about ⁠ ^ ⁠ is the following:

  7. Regular polygon - Wikipedia

    en.wikipedia.org/wiki/Regular_polygon

    If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of the polygon winds around the center m times. The (non-degenerate) regular stars of up to 12 sides are: Pentagram – {5/2} Heptagram – {7/2} and {7/3} Octagram – {8/3} Enneagram – {9/2} and {9/4} Decagram – {10/3}

  8. Descartes' theorem - Wikipedia

    en.wikipedia.org/wiki/Descartes'_theorem

    Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have.In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation.

  9. Area of a circle - Wikipedia

    en.wikipedia.org/wiki/Area_of_a_circle

    The same is true if we increase it to eight sides and so on. For a polygon with 2n sides, the parallelogram will have a base of length ns, and a height h. As the number of sides increases, the length of the parallelogram base approaches half the circle circumference, and its height approaches the circle radius.