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The radius of the incircle is related to the area of the triangle. [18] The ratio of the area of the incircle to the area of the triangle is less than or equal to π / 3 3 {\displaystyle \pi {\big /}3{\sqrt {3}}} , with equality holding only for equilateral triangles .
This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals , circumscribing quadrilaterals , and circumscriptible ...
In plane geometry, a mixtilinear incircle of a triangle is a circle which is tangent to two of its sides and internally tangent to its circumcircle. The mixtilinear incircle of a triangle tangent to the two sides containing vertex A {\displaystyle A} is called the A {\displaystyle A} -mixtilinear incircle.
The radius of the incircle of a right triangle with legs and and hypotenuse is = + = + +. The radius of the circumcircle is half the length of ...
where r is the incircle radius and R is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler's triangle inequality), with equality only in the equilateral case. [7] [8] The distance between O and the orthocenter H is [9] [10]
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius , and incenter and circumcenter respectively.
The incircle itself may be constructed by dropping a perpendicular from the incenter to one of the sides of the triangle and drawing a circle with that segment as its radius. [3] The incenter lies at equal distances from the three line segments forming the sides of the triangle, and also from the three lines containing those segments.
Euler's theorem: = | | = 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).