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 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

|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,

(3)

where

(4)

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 column (and taking a minus sign) and similarly for (this time taking the plus sign),

			(6)
			(7)

and 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

(9)

which is a circle of the form

(10)

with circumcenter

			(11)
			(12)

and circumradius

(13)

In exact trilinear coordinates , the equation of the circle passing through three noncollinear points with exact trilinear coordinates , , and 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 , , , ... 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.

triangle	circumcircle
anticomplementary triangle	anticomplementary circle
circum-medial triangle	circumcircle
circumnormal triangle	circumcircle
circum-orthic triangle	circumcircle
circumcircle mid-arc triangle	circumcircle
contact triangle	incircle
D-triangle	orthocentroidal circle
Euler-Gergonne-Soddy triangle	Euler-Gergonne-Soddy circle
Euler triangle	nine-point circle
excentral triangle	Bevan circle
extangents triangle	extangents circle
extouch triangle	Mandart circle
Feuerbach triangle	nine-point circle
first Brocard triangle	Brocard circle
first Morley triangle	Morley's circle
first Neuberg triangle	first Neuberg circle
Fuhrmann triangle	Fuhrmann circle
half-altitude triangle	half-altitude circle
hexyl triangle	hexyl circle
incentral triangle	incentral circle
inner Napoleon triangle	inner Napoleon circle
inner Vecten triangle	inner Vecten circle
intangents triangle	intangents circle
Lemoine triangle	third Lemoine circle
Lucas central triangle	Lucas central circle
Lucas inner triangle	Lucas inner triangle
Lucas tangents triangle	Lucas circles radical circle
medial triangle	nine-point circle
mid-arc triangle	incircle
mixtilinear triangle	mixtilinear circle
orthic triangle	nine-point circle
outer Napoleon triangle	outer Napoleon circle
outer Vecten triangle	outer Vecten circle
reference triangle	circumcircle
reflection triangle	reflection circle
second Brocard triangle	Brocard circle
second Neuberg triangle	second Neuberg circle
Stammler triangle	Stammler circle
Steiner triangle	second Steiner circle
symmedial triangle	symmedial circle
tangential mid-arc triangle	tangential mid-arc circle
tangential triangle	tangential circle
Yff central triangle	Yff central circle
Yff contact triangle	Yff contact circle
Yiu triangle	Yiu Circle

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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

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

Circumcircle

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