The

circumcircle of a triangle is the unique circle that passes through all three of its vertices.

Observe that each edge-crossing produced by moving the interior points from the incircle to the

circumcircle involves an edge connecting an interior point to its nearest vertex of the h-gon or involves an edge of the h-gon (the solid edges in Figure 6).

The upper bound is clear because pqr is enclosed by the

circumcircle with radius X.

Definition 9 We define the

circumcircle sphere as the sphere having the

circumcircle of the base triangle as diametral circle.

in an acute triangle ABC, where a,b,c are the lenghts of sides BC, CA, AB; A,B,C are the measures of the angles [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] calculated in radians, r is the radius of incircle; s is the semi-perimeter; R is the radius of

circumcircle and [DELTA] is the area.

i]) for l [element of] {i, j, k}, and the

circumcircle of ([p.

Geometry elements: point, glider, intersection, line, segment, parallel, perpendicular, axis, tangent, normal, vector, circle,

circumcircle, ellipse, hyperbola, parabola, conic defined by five points, polygon, regular polygon, midpoint, mirror point, reflection point, semicircle,

circumcircle arc,

circumcircle sector, angle, bisector, bisector lines, exact loci computation, homogeneous and affine coordinates

4 Some dynamic geometry software packages have the "custom tools" feature that allows users to construct their own tools such as the

Circumcircle of a Triangle, but general users have difficulties in constructing such "custom tools" themselves.

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

However, a geometric approach to the problem can be taken by observing that all the points D such that ADC = ABC lie on the

circumcircle of ABC.

n] tends to be an equilateral triangle with the same

circumcircle as the Napoleon's Triangle of the original triangle.

n] represent the radius of the

circumcircles of the test triangle and the source triangle, respectively.