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The center of the incircle is a triangle center called the triangle's incenter. [1] 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]
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:
A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon. A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.
The centers of the incircles and excircles form an orthocentric system. [26] The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle. [27] The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the ...
In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., y = 2x + 1 (a line), or x 2 + y 2 = 7 (a circle).
The triangle DEF is called the pedal triangle of P. [17] The antipedal triangle of P is the triangle formed by the lines through A, B, C perpendicular to PA, PB, PC respectively. Two points P and Q are called counter points if the pedal triangle of P is homothetic to the antipedal triangle of Q and the pedal triangle of Q is homothetic to the ...
The circle through A, C, and I has its center at D. In particular, this implies that the center of this circle lies on the circumcircle. [9] [10] The three triangles AID, CID, and ACD are isosceles, with D as their apex. A fourth point E, the excenter of ABC relative to B, also lies at the same distance from D, diametrically opposite from I. [5 ...
Illustration of Poncelet's porism for n = 3, a triangle that is inscribed in one circle and circumscribes another.. In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same ...