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An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
An angle bisector of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter, which is the center of the triangle's incircle. The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius.
The intersection of three circular disks forms a convex circular triangle. For instance, a Reuleaux triangle is a special case of this construction where the three disks are centered on the vertices of an equilateral triangle, with radius equal to the side length of the triangle. However, not every convex circular triangle is formed as an ...
The triangle's nine-point circle has half the diameter of the circumcircle. In any given triangle, the circumcenter is always collinear with the centroid and orthocenter. The line that passes through all of them is known as the Euler line. The isogonal conjugate of the circumcenter is the orthocenter.
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 ...
The Euler line of a triangle is a line passing through its circumcenter, centroid, and orthocenter, among other points. The incenter generally does not lie on the Euler line; [16] it is on the Euler line only for isosceles triangles, [17] for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle. Figure 4. The center N of the nine-point circle bisects a segment from the orthocenter H to the circumcenter O (making the orthocenter a center of dilation to both circles): [6]: p.152
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 is called the -mixtilinear incircle.