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• Isogonal conjugate
  '''Isogonal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...r of lines symmetric with respect to <math>\ell</math> and containing the point <math>O</math> be called isogonals with respect to the pair <math>(\ell,O).
  50 KB (8,743 words) - 18:35, 15 February 2025
• Barycentric coordinates
  ...iangle <math> \Delta{A_1}{A_2}{A_3} </math>. These masses then determine a point <math> P </math>, which is the geometric centroid of the three masses and i For any point in the plane <math>ABC</math> there are <i><b>barycentric coordinates(BC):
  34 KB (6,879 words) - 17:36, 2 February 2025
• Symmetry
  Let point <math>E</math> be symmetric <math>D</math> with respect <math>\ell.</math> [[File:Bisectors 2.png|350px|right]]
  19 KB (3,291 words) - 13:44, 6 October 2024
• 2022 USAMO Problems/Problem 2
  Let <math>b\geq2</math> and <math>w\geq2</math> be fixed integers, and <math>n=b+w</math>. Given are <math>2b</math> identical black [[File:Peppa-usamo-2022-2.png|800px]]
  7 KB (1,337 words) - 02:08, 9 January 2024
• Steiner line
  Let <math>ABC</math> be a triangle with orthocenter <math>H. P</math> is a point on the circumcircle <math>\Omega</math> of <math>\triangle ABC.</math> ...th>H</math> are collinear. Respective line is known as the Steiner line of point <math>P</math> with respect to <math>\triangle ABC.</math>
  14 KB (2,381 words) - 12:07, 12 May 2024
• Polya’s method for extremums
  ...<math>AB</math> has the ends on the sides of a right angle and contains a point <math>C(x_C, y_C).</math> Find the shortest length of such a segment. The rod is fixed at point <math>C</math> on a hinge without friction. The hinge allows the rod to rot
  12 KB (2,104 words) - 14:11, 24 February 2024
• DVI exam
  so the system of the equations define the only tangent point of the circle and the line. ...er, <math>O'</math> is the incenter <math>\triangle KK'L, F'</math> is the point of tangency incircle of <math>\triangle KK'L</math> and <math>KK'.</math>
  48 KB (8,214 words) - 08:59, 30 July 2024
• Projective geometry (simplest cases)
  ...pace. To each point <math>A</math> of plane <math>\Pi</math> we assign the point <math>A'</math> of plane <math>\Pi'</math> at which the line <math>AO</math ...ng which it intersects the plane <math>\Pi</math> as <math>\ell.</math> No point of the line <math>\ell</math> has an image in the plane <math>\Pi'.</math>
  40 KB (6,966 words) - 17:36, 20 March 2025
• Sharygin Olympiads, the best
  A point <math>D</math> lies inside a triangle <math>ABC</math> on the bisector of a ...ath>\angle PAB = \angle PBC = \angle PCA.</math> Prove that there exists a point <math>Q</math> on another median such that <math>\angle QBA = \angle QCB =
  49 KB (8,447 words) - 10:27, 21 March 2025