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• Spiral similarity
  ...math>A'B</math> to <math>AB'.</math> This fact explain existance of Miquel point. <math>C </math> is any point of <math>AB, \theta </math> is circle <math>CPx_0, C' = \theta \cap A'B'</m
  28 KB (4,687 words) - 22:27, 19 February 2025
• 2019 USAMO Problems/Problem 2
  ...of <math>ABCD</math> intersect at <math>E</math>. Let <math>P</math> be a point on side <math>\overline{AB}</math> satisfying <math>\angle APD = \angle BPC Note that only one point <math>P</math> satisfies the given angle condition. With this in mind, cons
  4 KB (620 words) - 22:51, 18 October 2022
• 2019 IMO Problems/Problem 2
  ...ath>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between [[File:2019 IMO 2a.png|420px|right]]
  3 KB (479 words) - 01:49, 19 November 2023
• 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