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• LaTeX:Symbols
  |<math>a^x</math>||a^x||<math>a^{xyz}</math>||a^{xyz}||<math>a_x</math>||a_x | <math>\omega</math>||\omega
  16 KB (2,315 words) - 19:35, 4 November 2024
• Euler line
  Let <math>\omega</math> be circumcircle of <math>\triangle ABC.</math> Let <math>\omega'</math> be the circle symmetric to <math>\omega</math> with respect to <math>AB.</math>
  59 KB (10,203 words) - 03:47, 30 August 2023
• Nine-point circle
  <math>\Omega = \odot ABC, Z</math> be the point on <math>\Omega</math> opposite <math>A.</math> ...AEHF</math> is a parallelogram. The line <math>EF</math> intersects <math>\Omega</math> at the points <math>X</math> and <math>Y.</math>
  6 KB (994 words) - 15:02, 12 March 2024
• 2007 AIME II Problems/Problem 15
  ...math>13,</math> <math>14,</math> and <math>15,</math> the radius of <math>\omega</math> can be represented in the form <math>\frac{m}{n}</math>, where <math ...= OO_C = 2r</math>, <math>O</math> is the circumcenter of <math>\triangle XYZ</math> and <math>\mathcal{H}</math> therefore maps the circumcenter of <mat
  11 KB (2,099 words) - 22:44, 6 October 2024
• Excircle
  </asy>|right|Triangle <math>\triangle XYZ</math> and its excircles.}} ...section of segments <math>AD_2</math> and <math>BE_2</math>. Circle <math>\omega</math> intersects segment <math>AD_2</math> at two points, the closer of wh
  5 KB (843 words) - 02:02, 1 July 2020
• 2009 AIME I Problems/Problem 12
  ...I}</math> and <math>\overline{BI}</math> are both tangent to circle <math>\omega</math>. The ratio of the perimeter of <math>\triangle ABI</math> to the len ...te that <math>AB=37</math>; let the tangents from <math>I</math> to <math>\omega</math> have length <math>x</math>. Then the perimeter of <math>\triangle AB
  13 KB (2,170 words) - 22:35, 3 September 2024
• 2013 USAJMO Problems
  ...ath>P</math> to line <math>XY</math>. Point <math>C</math> lies on <math>\omega</math> such that line <math>XC</math> is perpendicular to line <math>AZ</ma ...,y,z\geq 1</math> satisfying <cmath>\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.</cmath>
  3 KB (478 words) - 15:41, 5 August 2023
• 2016 AIME II Problems
  \log_2(xyz-3+\log_5 x)&=5,\\ \log_3(xyz-3+\log_5 y)&=4,\\
  8 KB (1,312 words) - 20:16, 3 March 2021
• 2006 iTest Problems
  ...49</math>. Find the maximum possible value for the product <math>xyz</math>. wxy^2 + wx^2z + w^2yz + xyz^2&=2\\
  31 KB (4,811 words) - 23:02, 3 November 2023
• 2016 USAMO Problems/Problem 3
  Let <math>\omega</math> be circumcircle of <math>\triangle ABC</math> centered at <math>O.</ Let <math>Y_1,</math> and <math>Z_1</math> be crosspoints of <math>\omega</math> and <math>BY,</math> and <math>CZ,</math> respectively.
  6 KB (998 words) - 20:36, 17 October 2022
• 2020 USOJMO Problems/Problem 2
  .../math>. Let <math>\ell</math> be a variable line that is tangent to <math>\omega</math> and meets the interior of segments <math>BC</math> and <math>CA</mat ...math> (This is trivial). <math>T</math> is the point of tangency of <math>\omega</math> and segment <math>\overline{PQ}</math>.
  7 KB (1,437 words) - 18:14, 6 October 2023
• 2022 AIME I Problems/Problem 8
  ...</math> and <math>\overline{AC}</math> and is internally tangent to <math>\omega.</math> Circles <math>\omega_B</math> and <math>\omega_C</math> are defined dot("$\omega$",W,1.5*dir(270),linewidth(4));
  12 KB (1,955 words) - 20:11, 31 January 2024
• 2004 Pan African MO Problems/Problem 6
  ...<math>XYZ</math> for any three points <math>X, Y, Z</math>, and use <math>\Omega</math> to represent the circumcircle of <math>ABCD</math>. Without loss of Now, the [[radical center]] of <math>\omega_{ADN'}, \omega_{BCN'}, \Omega</math> must be <math>J</math>, so that the [[radical axis]] of <math>\omega
  3 KB (572 words) - 12:48, 27 May 2024
• 2021 AMC 10A Problems/Problem 13
  We will place tetrahedron <math>ABCD</math> in the <math>xyz</math>-plane. By the Converse of the Pythagorean Theorem, we know that <mat == Video Solution by Omega Learn (Using Pythagorean Theorem, 3D Geometry: Tetrahedron) ==
  3 KB (496 words) - 19:22, 10 November 2024
• 2021 WSMO Speed Round Problems
  ...Z</math>, and <math>D</math> is on <math>XY</math>. Find the area of <math>XYZ</math>. ...wn which is tangent to <math>AB,AC</math> and externally tangent to <math>\omega</math>. The radius of <math>\omega_2</math> can be expressed as <math>\frac
  4 KB (707 words) - 11:38, 6 June 2022
• Spieker center
  Let <math>\triangle ABC</math> be given. Let <math>\omega, \omega_A, \omega_B, \omega_C</math> be incircle, A-excircle, B-excircle, C The distances from <math>M_A</math> to the tangent points of <math>\omega</math> and <math>\omega_A</math> are the same, so <math>M_A \in r_A.</math>
  8 KB (1,499 words) - 05:48, 8 August 2023
• Arcticturn Prep
  ...h <math>AP<AQ</math>. Rays <math>CP</math> and <math>CQ</math> meet <math>\omega</math> again at <math>S</math> and <math>T</math> (other than <math>C</math ...ations <math>x^3 - xyz = 2</math>, <math>y^3 - xyz = 6</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> ca
  18 KB (3,089 words) - 10:53, 27 August 2024