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• 2002 AIME I Problems/Problem 10
  In the diagram below, angle <math>ABC</math> is a right angle. Point <math>D</math> is on <math>\overline{BC}</math>, and <ma ...> is <math>10/3</math> that of triangle <math>AEG</math>, since they share a common side and angle, so the area of triangle <math>AGF</math> is <math>10
  4 KB (643 words) - 22:44, 8 August 2023
• 2003 AIME II Problems/Problem 11
  Triangle <math>ABC</math> is a right triangle with <math>AC = 7,</math> <math>BC = 24,</math> and right an ...>, we get <math>DM=\frac{5\sqrt{11}} {2}</math>. Since <math>ABC</math> is a right triangle, <math>M</math> is the circumcenter and thus, <math>CM=\frac
  5 KB (772 words) - 19:47, 1 August 2023
• 2002 AIME II Problems/Problem 14
  ...is <math>152</math>, and the angle <math>PAM</math> is a [[right angle]]. A [[circle]] of [[radius]] <math>19</math> with center <math>O</math> on <mat ...MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in similar tr
  4 KB (658 words) - 19:15, 19 December 2021
• 2001 AIME II Problems/Problem 13
  pair A=(0,0), D=(x,0), B=IP(CR(A,8),CR(D,10)), E=(-3x/5,0), C=IP(CR(E,16),CR(D,64/5)); D(MP("A",A)--MP("B",B,NW)--MP("C",C,NW)--MP("D",D)--cycle); D(B--D); D(A--MP("E",E)--B,d);
  4 KB (743 words) - 03:32, 23 January 2023
• 2006 USAMO Problems/Problem 6
  ...th>, <math>SBF</math>, <math>TCF</math>, and <math>TDE</math> pass through a common point. ...imilarity, and <math>X</math> is the center of spiral similarity for <math>A,E,B,</math> and <math>F</math>.
  5 KB (986 words) - 22:46, 18 May 2015
• Dedekind domain
  A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * <math>R</math> is a [[noetherian]] [[ring]].
  9 KB (1,648 words) - 16:36, 14 October 2017
• 1959 IMO Problems/Problem 6
  ...</math> (with <math>AB </math> [[parallel]] to <math>DC </math>) in which a [[circle]] can be inscribed, and with [[vertex | vertices]] <math>B </math> ...that we must have both lines <math>AB </math> (which we shall denote <math>a </math>) and <math>DC </math> (which we shall denote <math>c </math>) paral
  2 KB (383 words) - 05:58, 11 February 2024
• 2006 Romanian NMO Problems/Grade 8/Problem 1
  We consider a prism with 6 faces, 5 of which are circumscriptible quadrilaterals. Prove t ...o non-coplanar circles in space that intersect at two points, there exists a point P such that P is equidistant from any point on one circle and any poi
  3 KB (509 words) - 23:22, 15 August 2012
• 2006 Romanian NMO Problems/Grade 9/Problem 2
  ...ath>\triangle BDC</math>. Now assume that P' is the point such that DP' is a median of <math>\triangle EDF</math> (it is on <math>EF</math>). It is on D
  3 KB (488 words) - 14:05, 15 December 2022
• Equilateral triangle
  pair A=(0,50),B=(43.301,-25),C=(-43.301,-25); draw(A--B--C--A);
  1 KB (186 words) - 19:57, 15 September 2022
• Similarity (geometry)
  ...similar in every aspect except possibly size or orientation. For example, a globe and the surface of the earth are, in theory, similar. ...id motion]]s). We say two objects are similar if they are congruent up to a [[dilation]].
  2 KB (261 words) - 20:42, 25 November 2023
• Euler line
  ...] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain fixed orders and distan ...followed by a homothety with scale factor <math>2</math> centered at <math>A</math> brings <math>\triangle ABC \to \triangle O_AO_BO_C</math>. Let us ex
  59 KB (10,203 words) - 04:47, 30 August 2023
• 2003 AMC 10A Problems/Problem 22
  pair D=(0,0), Ep=(4,0), A=(9,0), B=(9,8), H=(3,8), C=(0,8), G=(-6,20), F=(-6,0); draw(D--A--B--C--D--F--G--Ep);
  9 KB (1,446 words) - 22:48, 8 May 2024
• 2006 Canadian MO Problems/Problem 2
  Let <math>ABC</math> be an [[acute angle]]d [[triangle]]. Inscribe a [[rectangle]] <math>DEFG</math> in this triangle so that <math>D</math> is ...}{AB}</math> and let <math>H</math> be the foot of the altitude from <math>A</math> to <math>BC</math>. Then by [[similarity]], <math>\frac{AE}{AC} = \
  2 KB (416 words) - 20:00, 21 September 2014
• Mock AIME 3 Pre 2005 Problems/Problem 11
  ...th> is an acute triangle with perimeter <math>60</math>. <math>D</math> is a point on <math>\overline{BC}</math>. The circumcircles of triangles <math>A ...ath>B</math> to <math>E</math> and <math>F</math> to <math>D</math>, so by a fundamental theorem of spiral similarity <math>\triangle BDE\sim\triangle F
  2 KB (325 words) - 15:32, 22 March 2015
• Circumradius
  ...very triangle is [[cyclic]], every triangle has a circumscribed circle, or a [[circumcircle]]. ==Formula for a Triangle==
  4 KB (729 words) - 16:52, 19 February 2024
• 2006 AIME II Problems/Problem 6
  ...d <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> pair B = (0, 0), C = (1, 0), D = (1, 1), A = (0, 1);
  7 KB (1,067 words) - 12:23, 8 April 2024
• 1961 IMO Problems
  \quad x + y + z \!\!\! &= a \; \, \\ ...</math> and <math>b </math> are constants. Give the conditions that <math>a </math> and <math>b </math> must satisfy so that <math>x, y, z </math> (the
  3 KB (425 words) - 21:18, 20 August 2020
• 1961 IMO Problems/Problem 6
  ...o not form a triangle.) What is the locus of point <math>G</math> as <math>A', B', C'</math> range independently over the plane <math>\epsilon</math>? ...\prime B^\prime C^\prime</math>. It is obvious that the centroid of <math>A^\prime B^\prime C^\prime</math> can be any point on plane <math>\epsilon</m
  2 KB (301 words) - 23:29, 18 July 2016
• 1962 IMO Problems
  (a) Its decimal representation has 6 as the last digit. ...> begin their motion at the same instant from the starting positions <math>A</math> and <math>B'</math>, respectively. Determine and draw the locus of t
  3 KB (431 words) - 21:17, 20 August 2020

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