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  2. Incircle and excircles - Wikipedia

    en.wikipedia.org/wiki/Incircle_and_excircles

    An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.

  3. Circular triangle - Wikipedia

    en.wikipedia.org/wiki/Circular_triangle

    Circular triangles give the solution to an isoperimetric problem in which one seeks a curve of minimum length that encloses three given points and has a prescribed area. . When the area is at least as large as the circumcircle of the points, the solution is any circle of that area surrounding the poi

  4. Circumcircle - Wikipedia

    en.wikipedia.org/wiki/Circumcircle

    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.

  5. Circle - Wikipedia

    en.wikipedia.org/wiki/Circle

    A circle is a shape consisting of all ... the Pythagorean theorem can be used to calculate the radius of the unique circle that ... About every triangle a unique ...

  6. Spherical trigonometry - Wikipedia

    en.wikipedia.org/wiki/Spherical_trigonometry

    Construct the great circle from A that is normal to the side BC at the point D. Use Napier's rules to solve the triangle ABD: use c and B to find the sides AD and BD and the angle ∠BAD. Then use Napier's rules to solve the triangle ACD: that is use AD and b to find the side DC and the angles C and ∠DAC. The angle A and side a follow by ...

  7. Thales's theorem - Wikipedia

    en.wikipedia.org/wiki/Thales's_theorem

    Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle.. In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle.

  8. Conway circle theorem - Wikipedia

    en.wikipedia.org/wiki/Conway_circle_theorem

    In plane geometry, the Conway circle theorem states that when the sides meeting at each vertex of a triangle are extended by the length of the opposite side, the six endpoints of the three resulting line segments lie on a circle whose centre is the incentre of the triangle. The circle on which these six points lie is called the Conway circle of ...

  9. Nine-point circle - Wikipedia

    en.wikipedia.org/wiki/Nine-point_circle

    The nine-point circle is tangent to the incircle and excircles. In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that: