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The MacBeath circle, a term coined here for the first time, is the circumcircle of the MacBeath triangle. It has a fairly complicated radius, center function, and circle ...

The mixtilinear circle is the circumcircle of the mixtilinear triangle, i.e., the triangle formed by the centers of the mixtilinear incircles. Neither its center not circle ...

The radical circle of the mixtilinear incircles has a center with trilinear center function alpha_(999)=cosA-2, (1) which is Kimberling center X_(999). It has radius (2) ...

Morley's circle is the circumcircle of the first Morley triangle. Its center is the first Morley center, which has center function alpha_(356)=cos(1/3A)+2cos(1/3B)cos(1/3C), ...

The orthoptic circle of the Steiner inellipse is the circle with center at alpha_2=1/a, (1) corresponding to the triangle centroid G and radius ...

The outer Vecten circle is the circumcircle of the outer Vecten triangle. It has center at Kimberling center X_(641), which is the complement of the outer Vecten point ...

If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal. ...

Let t, u, and v be the lengths of the tangents to a circle C from the vertices of a triangle with sides of lengths a, b, and c. Then the condition that C is tangent to the ...

The distance from the center of a circle to its perimeter, or from the center of a sphere to its surface. The radius is equal to half the diameter.

The second Neuberg circle is the circumcircle of the second Neuberg triangle. The center has center function which is not a Kimberling center. Its radius is slightly ...