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In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter .
Given a right triangle ABC with hypotenuse AC, construct a circle Ω whose diameter is AC. Let O be the center of Ω. Let D be the intersection of Ω and the ray OB. By Thales's theorem, ∠ ADC is right. But then D must equal B. (If D lies inside ABC, ∠ ADC would be obtuse, and if D lies outside ABC, ∠ ADC would be acute.)
Malfatti's assumption that the two problems are equivalent is incorrect. Lob and Richmond (), who went back to the original Italian text, observed that for some triangles a larger area can be achieved by a greedy algorithm that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within one of the three remaining corners of the triangle, the one with the ...
Ptolemy's Theorem yields as a corollary a pretty theorem [2] regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle and a point on the circle. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle. [19] About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices. [20]
Inscribed circles of various polygons An inscribed triangle of a circle A tetrahedron (red) inscribed in a cube (yellow) which is, in turn, inscribed in a rhombic triacontahedron (grey). (Click here for rotating model) In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or ...
Construction of a tangent to a given circle (black) from a given exterior point (P). Thales' theorem may be used to construct the tangent lines to a point P external to the circle C: A circle is drawn centered on the midpoint of the line segment OP, having diameter OP, where O is again the center of the circle C.
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).