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 of the circumcircle is called the circumcenter, and the circle's radius is called the circumradius. A triangle's three perpendicular bisectors , , and meet (Casey 1888, p. 9) at (Durell 1928). The Steiner point and Tarry point lie on the circumcircle.
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
(1)
(Kimberling 1998, pp. 39 and 219). Extending the list of Kimberling (1998, p. 228), the circumcircle passes through the Kimberling centers for , 98 (Tarry point), 99 (Steiner point), 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110 (focus of the Kiepert parabola), 111 (Parry point), 112, 476 (Tixier point), 477, 675, 681, 689, 691, 697, 699, 701, 703, 705, 707, 709, 711, 713, 715, 717, 719, 721, 723, 725, 727, 729, 731, 733, 735, 737, 739, 741, 743, 745, 747, 753, 755, 759, 761, 767, 769, 773, 777, 779, 781, 783, 785, 787, 789, 791, 793, 795, 797, 803, 805, 807, 809, 813, 815, 817, 819, 825, 827, 831, 833, 835, 839, 840, 841, 842, 843, 898, 901, 907, 915, 917, 919, 925, 927, 929, 930, 931, 932, 933, 934, 935, 953, 972, 1113, 1114, 1141 (Gibert point), 1286, 1287, 1288, 1289, 1290, 1291, 1292, 1293, 1294, 1295, 1296, 1297, 1298, 1299, 1300, 1301, 1302, 1303, 1304, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1379, 1380, 1381, 1382, 1477, 2222, 2249, 2291, 2365, 2366, 2367, 2368, 2369, 2370, 2371, 2372, 2373, 2374, 2375, 2376, 2377, 2378, 2379, 2380, 2381, 2382, 2383, 2384, 2687, 2688, 2689, 2690, 2691, 2692, 2693, 2694, 2695, 2696, 2697, 2698, 2699, 2700, 2701, 2702, 2703, 2704, 2705, 2706, 2707, 2708, 2709, 2710, 2711, 2712, 2713, 2714, 2715, 2716, 2717, 2718, 2719, 2720, 2721, 2722, 2723, 2724, 2725, 2726, 2727, 2728, 2729, 2730, 2731, 2732, 2733, 2734, 2735, 2736, 2737, 2738, 2739, 2740, 2741, 2742, 2743, 2744, 2745, 2746, 2747, 2748, 2749, 2750, 2751, 2752, 2753, 2754, 2755, 2756, 2757, 2758, 2759, 2760, 2761, 2762, 2763, 2764, 2765, 2766, 2767, 2768, 2769, 2770, 2855, 2856, 2857, 2858, 2859, 2860, 2861, 2862, 2863, 2864, 2865, 2866, 2867, and 2868.
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 is taken on the circumcircle, then the feet , , and of the perpendiculars from to the sides (or their extensions) of the triangle are collinear on a line called the Simson line. Furthermore, the reflections , , of any point on the circumcircle taken with respect to the sides , , of the triangle are collinear, not only with each other but also with the orthocenter (Honsberger 1995, pp. 44-47).
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 for , 2, 3 is
(2)
Expanding the determinant,
(3)
where
(4)
is the determinant obtained from the matrix
(5)
by discarding the column (and taking a minus sign) and similarly for (this time taking the plus sign),
and is given by
(8)
Completing the square gives
(9)
which is a circle of the form
(10)
with circumcenter
and circumradius
(13)
In exact trilinear coordinates , the equation of the circle passing through three noncollinear points with exact trilinear coordinates , , and is
(14)
(Kimberling 1998, p. 222).
If a polygon with side lengths , , , ... and standard trilinear equations , , , ... has a circumcircle, then for any point of the circle,
(15)
(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 Point Explore with Wolfram|Alpha 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: 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: 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: 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: Macmillian, pp. 66-70, 1893.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995. Referenced on Wolfram|AlphaCircumcircle Cite this as:Weisstein, Eric W. "Circumcircle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Circumcircle.html
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