Since this equicircle lays outside the triangle, it is named an escribed circle, or excircle.
Point D is where the excircle touches the shorter side of the triangle.
Let the excircle
of triangle ABC opposite the vertex A be tangent to the side BC at the point [A.
Table 2-1] Other centers of tetrahedron Position in Position in Centers 2D Geometry 3D Geometry Circumcenter A point where three A point where perpendicular bisectors perpendicular bisecting intersect planes intersect Centroid A point where three A point where median medians intersect planes (Planes with a edge and its opposite edge's middle point) intersect Excenter A point where exterior A point where exterior angle bisectors intersect dihedral-bisecting planes intersect Centers Property Circumcenter Becomes the center of the circumcircle and the circumsphere, respectively Centroid Divides the line which connects a point and the opposite planes' centroid as 2:1, 3:1 respectively Excenter Becomes the center of the excircle
and the exosphere, respectively
Moreover, there are four congruent right triangles in the rectangle ABCD, and each one is matched with a different circle in the dual system, showing it to be an excircle or incircle.
Mack and Czernezkyj carry out a geometric construction of the Pythagorean Tree using each excircle.
In fact, denoting the diameter of the excircle on the hypotenuse by w, then:
and, since the RHS is always even for PPTs, it follows that the radius of this excircle is also an integer.
In addition, we have the result that, if (a, b, c) is a PPT, and if w = x = a + b + c is the diameter of the excircle on the hypotenuse, c, then (2w - b, 2w - a, 2w + c) is also a PPT.
The Nagel point is the point of intersection of the line segments from the vertices of the triangle to the points of tangency of the opposite excircles .
The Nagel point N of a triangle ABC is defined as the point of intersection of the cevians AA, BB', and CC', where A', F, and C' are points where the excircles touch the sides (Figure 1).
The cevians joining the vertices of a triangle to the points of tangency of the opposite sides with the corresponding excircles are concurrent.