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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.)
Not every polygon with more than three sides is an inscribed polygon of a circle; those polygons that are so inscribed are called cyclic polygons. Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangle's centroid. Every triangle has an infinitude of inscribed ...
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.
It is a theorem in Euclidean geometry that the three interior angle bisectors of a triangle meet in a single point. In Euclid's Elements, Proposition 4 of Book IV proves that this point is also the center of the inscribed circle of the triangle. The incircle itself may be constructed by dropping a perpendicular from the incenter to one of the ...
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] A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon. [21]
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 ...
The large triangle that is inscribed in the circle gets subdivided into three smaller triangles, all of which are isosceles because their upper two sides are radii of the circle. Inside each isosceles triangle the pair of base angles are equal to each other, and are half of 180° minus the apex angle at the circle's center.