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Circumcircle -- from Wolfram MathWorld
Circumcircle
DOWNLOAD Mathematica NotebookCircumcircle
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.
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.
SimsonLineCircumcircleOrthoLine
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)