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GeezerVison is showing mathworld.wolfram.com — Circumcircle -- from Wolfram MathWorld 
Circumcircle
The circumcircle is a triangle's circumscribed circle, i.e., the unique circle that passes through each of the
triangle's three vertices. The center A circumcircle of a polygon is the two-dimensional case of a circumsphere of a solid. The circumcircle can be specified using trilinear coordinates as
(Kimberling 1998, pp. 39 and 219). Extending the list of Kimberling (1998, p. 228), the circumcircle passes through the Kimberling centers It is orthogonal to the Parry circle and Stevanović circle. The polar triangle of the circumcircle is the tangential triangle. The circumcircle is the anticomplement of the nine-point circle.
When an arbitrary point The tangent to a triangle's circumcircle at a vertex is antiparallel to the opposite side, the sides of the orthic triangle are parallel to the tangents to the circumcircle at the vertices, and the radius of the circumcircle at a vertex is perpendicular to all lines antiparallel to the opposite sides (Johnson 1929, pp. 172-173).
A geometric construction for the circumcircle is given by Pedoe (1995, pp. xii-xiii). The equation for the circumcircle of
the triangle with polygon
vertices
Expanding the determinant,
where
by discarding the
and
Completing the square gives
which is a circle of the form
with circumcenter
and circumradius
In exact trilinear coordinates
(Kimberling 1998, p. 222).
If a polygon with side lengths
(Casey 1878, 1893). The following table summarizes named circumcircles of a number of named triangles. See alsoCevian Circle, Circle, Circumcenter, Circumradius, Circumsphere, Enclosing Circle, Excircles, Incircle, Minimal Enclosing Circle, Parry Point, Pivot Theorem, Purser's Theorem, Simson Line, Steiner Points, Tarry PointExplore with Wolfram|Alpha
More things to try: ReferencesCasey, J. "On the Equations of Circles (Second Memoir)." Trans. Roy. Irish Acad. 26, 527-610, 1878.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., 1888.Casey, J. A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin, Ireland: Hodges, Figgis, & Co., pp. 128-129, 1893.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 7, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London, England: Macmillan, pp. 19-20, 1928.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Lachlan, R. "The Circumcircle." §118-122 in An Elementary Treatise on Modern Pure Geometry. London, England: Macmillian, pp. 66-70, 1893.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.Referenced on Wolfram|AlphaCircumcircleCite this as:Weisstein, Eric W. "Circumcircle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Circumcircle.html Subject classifications | ||||||||||||||||||||||||||||||||||||||||
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of the circumcircle is called the 
is called the 
,
,
meet (Casey 1888, p. 9) at 
(Durell 1928). The 
and 
lie on the circumcircle.
for 
,
is taken on the circumcircle, then the feet 
,
,
of the perpendiculars from 
to the sides (or their extensions) of the 
,
,
of any point 
on the circumcircle taken with respect to the sides 
,
,
of the triangle are 
(Honsberger 1995, pp. 44-47).
for 
,
is the determinant obtained from the
matrix
![D=[x_1^2+y_1^2 x_1 y_1 1; x_2^2+y_2^2 x_2 y_2 1; x_3^2+y_3^2 x_3 y_3 1]](/api/image?u=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FCircumcircle%2FNumberedEquation5.svg)
column (and taking a minus sign) and similarly for 
(this time taking the plus sign),
is given by
,
,
,
is
,
,
,
,
,
,