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Circumcircle


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 O of the circumcircle is called the circumcenter, and the circle's radius R is called the circumradius. A triangle's three perpendicular bisectors M_A, M_B, and M_C meet (Casey 1888, p. 9) at O (Durell 1928). The Steiner point S and Tarry point T 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

 abetagamma+bgammaalpha+calphabeta=0
(1)

(Kimberling 1998, pp. 39 and 219). Extending the list of Kimberling (1998, p. 228), the circumcircle passes through the Kimberling centers X_i for i=74, 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.

SimsonLine
CircumcircleOrthoLine

When an arbitrary point P is taken on the circumcircle, then the feet P_1, P_2, and P_3 of the perpendiculars from P to the sides (or their extensions) of the triangle are collinear on a line called the Simson line. Furthermore, the reflections P_A, P_B, P_C of any point P on the circumcircle taken with respect to the sides BC, AC, AB of the triangle are collinear, not only with each other but also with the orthocenter H (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 (x_i,y_i) for i=1, 2, 3 is

 |x^2+y^2 x y 1; 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|=0.
(2)

Expanding the determinant,

 a(x^2+y^2)+b_xx+b_yy+c=0,
(3)

where

 a=|x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|,
(4)

b_x 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]
(5)

by discarding the x_i column (and taking a minus sign) and similarly for b_y (this time taking the plus sign),

b_x=-|x_1^2+y_1^2 y_1 1; x_2^2+y_2^2 y_2 1; x_3^2+y_3^2 y_3 1|
(6)
b_y=|x_1^2+y_1^2 x_1 1; x_2^2+y_2^2 x_2 1; x_3^2+y_3^2 x_3 1|,
(7)

and c is given by

 c=-|x_1^2+y_1^2 x_1 y_1; x_2^2+y_2^2 x_2 y_2; x_3^2+y_3^2 x_3 y_3|.
(8)

Completing the square gives

 a(x+(b_x)/(2a))^2+a(y+(b_y)/(2a))^2-(b_x^2)/(4a)-(b_y^2)/(4a)+c=0
(9)

which is a circle of the form

 (x-x_0)^2+(y-y_0)^2=r^2,
(10)

with circumcenter

x_0=-(b_x)/(2a)
(11)
y_0=-(b_y)/(2a)
(12)

and circumradius

 r=(sqrt(b_x^2+b_y^2-4ac))/(2|a|).
(13)

In exact trilinear coordinates (alpha,beta,gamma), the equation of the circle passing through three noncollinear points with exact trilinear coordinates (alpha_1,beta_1,gamma_1), (alpha_2,beta_2,gamma_2), and (alpha_3,beta_3,gamma_3) is

 |abetagamma+bgammaalpha+calphabeta alpha beta gamma; abeta_1gamma_1+bgamma_1alpha_1+calpha_1beta_1 alpha_1 beta_1 gamma_1; abeta_2gamma_2+bgamma_2alpha_2+calpha_2beta_2 alpha_2 beta_2 gamma_2; abeta_3gamma_3+bgamma_3alpha_3+calpha_3beta_3 alpha_3 beta_3 gamma_3|=0
(14)

(Kimberling 1998, p. 222).

If a polygon with side lengths a, b, c, ... and standard trilinear equations alpha=0, beta=0, gamma=0, ... has a circumcircle, then for any point of the circle,

 a/alpha+b/beta+c/gamma+...=0
(15)

(Casey 1878, 1893).

The following table summarizes named circumcircles of a number of named triangles.

trianglecircumcircle
anticomplementary triangleanticomplementary circle
circum-medial trianglecircumcircle
circumnormal trianglecircumcircle
circum-orthic trianglecircumcircle
circumcircle mid-arc trianglecircumcircle
contact triangleincircle
D-triangleorthocentroidal circle
Euler-Gergonne-Soddy triangleEuler-Gergonne-Soddy circle
Euler trianglenine-point circle
excentral triangleBevan circle
extangents triangleextangents circle
extouch triangleMandart circle
Feuerbach trianglenine-point circle
first Brocard triangleBrocard circle
first Morley triangleMorley's circle
first Neuberg trianglefirst Neuberg circle
Fuhrmann triangleFuhrmann circle
half-altitude trianglehalf-altitude circle
hexyl trianglehexyl circle
incentral triangleincentral circle
inner Napoleon triangleinner Napoleon circle
inner Vecten triangleinner Vecten circle
intangents triangleintangents circle
Lemoine trianglethird Lemoine circle
Lucas central triangleLucas central circle
Lucas inner triangleLucas inner triangle
Lucas tangents triangleLucas circles radical circle
medial trianglenine-point circle
mid-arc triangleincircle
mixtilinear trianglemixtilinear circle
orthic trianglenine-point circle
outer Napoleon triangleouter Napoleon circle
outer Vecten triangleouter Vecten circle
reference trianglecircumcircle
reflection trianglereflection circle
second Brocard triangleBrocard circle
second Neuberg trianglesecond Neuberg circle
Stammler triangleStammler circle
Steiner trianglesecond Steiner circle
symmedial trianglesymmedial circle
tangential mid-arc triangletangential mid-arc circle
tangential triangletangential circle
Yff central triangleYff central circle
Yff contact triangleYff contact circle
Yiu triangleYiu Circle

See also

Cevian 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

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References

Casey, 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.

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Circumcircle

Cite this as:

Weisstein, Eric W. "Circumcircle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Circumcircle.html

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