Circumcircle of a Triangle

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Mathematics Geometry Circumcircle of a Triangle

	Circumcircle of a Triangle
	The circumcircle of a triangle is a circle that passes through all of the vertices of the triangle. The circumcircle of a polygon is a circle that passes through all of the vertices of the polygon. The circumcenter of a triangle is the center of the circumcircle of the triangle. The circumcenter's position depends on the type of triangle • If and only if a triangle is acute (all angles smaller than a right angle), the circumcenter lies inside the triangle • If and only if it is obtuse (has one angle bigger than a right angle), the circumcenter lies outside • If and only if it is a right triangle, the circumcenter lies on one of its sides (namely, the hypotenuse). This is one form of Thales' theorem. Inscribed Circle/Incircle: The incircle is the inscribed circle of a triangle ABC, i.e., the unique circle that is tangent to each of the triangle's three sides. The center I of the incircle is called the incenter, and the radius r of the circle is called the inradius. The incenter is the point of concurrence of the triangle's angle bisectors. In addition, the points MA, MB and MC of intersection of the incircle with the sides of triangle ABC are the polygon vertices of the pedal triangle taking the incenter as the pedal point (c.f. tangential triangle). This triangle is called the contact triangle. Area of Parallelogram and Triangles