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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 center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. [3] [4] The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two.
from the formula for the tangent of the difference of angles. Using s instead of r in the above formulas will give the same primitive Pythagorean triple but with a and b swapped. Note that r and s can be reconstructed from a, b, and c using r = a / (b + c) and s = b / (a + c).
All triangles can have an incircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be tangential is a non-square rectangle. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to be able to have an incircle.
Familiar examples of inscribed figures include circles inscribed in triangles or regular polygons, and triangles or regular polygons inscribed in circles. A circle inscribed in any polygon is called its incircle, in which case the polygon is said to be a tangential polygon.
Let I be the center of the incircle of triangle ABC, r its radius and F a, F b and F c the three points where the incircle touches the triangle sides a, b and c. Since the (extended) triangle sides are tangents of the incircle it follows that IF a, IF b and IF c are perpendicular to a, b and c.
Each of the three circles centered at the vertices crosses two sides of the triangle at right angles, at one of the three intouch points of the triangle, where its incircle is tangent to the side. The two circles tangent to these three circles are separated by the incircle, one interior to it and one exterior.
Feuerbach's theorem: the nine-point circle is tangent to the incircle and excircles of a triangle. The incircle tangency is the Feuerbach point. In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle.