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• Cyclic quadrilateral
  ...[triangles]] are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties. In a quadrilateral <math>ABCD</math>:
  1 KB (162 words) - 19:39, 9 March 2024
• 1988 AJHSME Problems/Problem 5
  If <math>\angle \text{CBD}</math> is a [[right angle]], then this protractor indicates that the measure of <math>\ pair A,B,C,D;
  1 KB (160 words) - 15:53, 17 December 2020
• 2005 AMC 12B Problems/Problem 6
  ...point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>? \textbf{(A) }\ 3 \qquad
  2 KB (299 words) - 14:29, 5 July 2022
• 1994 AIME Problems/Problem 10
  In triangle <math>ABC,\,</math> angle <math>C</math> is a right angle and the altitude from <math>C\,</math> meets <math>\overline{AB Since <math>\triangle ABC \sim \triangle CBD</math>, we have <math>\frac{BC}{AB} = \frac{29^3}{BC} \Longrightarrow BC^2
  3 KB (536 words) - 08:54, 5 November 2024
• 2003 AIME I Problems/Problem 15
  ...name of a point represent the mass located there. Since we are looking for a ratio, we assume that <math>AB=120</math>, <math>BC=169</math>, and <math>C First, reflect point <math>F</math> over angle bisector <math>BD</math> to a point <math>F'</math>.
  8 KB (1,382 words) - 23:37, 11 July 2024
• 2000 AIME II Problems/Problem 8
  ...ath>AC \perp BD</math>, it follows that <math>\triangle BAC \sim \triangle CBD</math>, so <math>\frac{x}{\sqrt{11}} = \frac{y}{x} \Longrightarrow x^2 = y\ ...th>\overline{CD}</math>. Then <math>AE = x</math>, and <math>ADE</math> is a [[right triangle]]. By the [[Pythagorean Theorem]],
  4 KB (584 words) - 18:35, 7 December 2019
• Mock AIME 1 2006-2007 Problems/Problem 7
  ...lic, since <math>\angle AEB=\angle AFB</math>. We then have, from Power of a Point, that <math>CE\cdot CA=CF\cdot CB</math>. In other words, <math>1\cdo ...{DB}{AB}\cdot [ABC]=\frac{x}{9}</math> and <math>[CFD]=\frac{CF}{CB}\cdot [CBD]=\frac{2}{3}\cdot \frac{x}{9}=\frac{2x}{27}</math>. The desired ratio is th
  3 KB (518 words) - 15:54, 25 November 2015
• Angle
  An '''angle''' is the [[union]] of two [[ray]]s with a common [[endpoint]]. The common endpoint of the rays is called the [[verte ...most common form is <math>\angle ABC</math>, read "angle ABC", where <math>A,C</math> are points on the sides of the angle and <math>B</math> is the ver
  4 KB (597 words) - 17:39, 9 May 2024
• 2001 IMO Problems/Problem 1
  ...at <math>\angle C \geq \angle B+30^{\circ}</math>. Prove that <math>\angle A+\angle COP < 90^{\circ}</math>. ...of the circumcircle. But <math>AZ = YP</math> (since <math>AZYP</math> is a rectangle).
  2 KB (417 words) - 16:24, 21 July 2018
• 2007 BMO Problems/Problem 1
  Let <math>ABCD </math> be a convex quadrilateral with <math>AB=BC=CD </math>, <math>AC \neq BD </math>, ...ath>, <math> \angle BAC = \angle ACB </math>, and similarly, <math> \angle CBD = \angle BDC </math>. Since <math> \angle CEB = \angle AED </math>, by con
  3 KB (566 words) - 22:59, 14 September 2014
• 2008 AMC 12A Problems/Problem 20
  <math>\mathrm{(A)}\ \frac{1}{28}\left(10-\sqrt{2}\right)\qquad\mathrm{(B)}\ \frac{3}{56}\lef pair C=(0,0),A=(0,3),B=(4,0),D=(4-2.28571,1.71429);
  6 KB (951 words) - 15:31, 2 August 2019
• 2008 AMC 10B Problems/Problem 24
  <math>\mathrm{(A)}\ 75\qquad\mathrm{(B)}\ 80\qquad\mathrm{(C)}\ 85\qquad\mathrm{(D)}\ 90\qqu ...solution requires the use of cyclic quadrilateral properties but could be a bit time-consuming during the contest.
  12 KB (1,944 words) - 17:31, 28 August 2024
• Spiral similarity
  ...milarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the com Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same o
  28 KB (4,853 words) - 22:23, 19 November 2024
• 2009 AMC 10A Problems/Problem 10
  Triangle <math>ABC</math> has a right angle at <math>B</math>. Point <math>D</math> is the foot of the alti pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
  5 KB (886 words) - 09:38, 16 November 2024
• 1988 AJHSME Problems
  The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of <math>\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)
  14 KB (1,872 words) - 14:23, 17 January 2023
• 2009 AIME I Problems/Problem 12
  ...cle having <math>\overline{CD}</math> as a diameter. Let <math>I</math> be a point outside <math>\triangle ABC</math> such that <math>\overline{AI}</mat ...s very complex but we can take a guess here. Assume that <math>x</math> is a rational number
  13 KB (2,170 words) - 22:35, 3 September 2024
• 2009 USAMO Problems
  ...th>S</math>. Prove that if <math>P, Q, R</math> and <math>S</math> lie on a circle then the center of this circle lies on line <math>XY</math>. ...elements <math>a, b, c</math> (not necessarily distinct) satisfying <math>a + b + c = 0</math>.
  4 KB (718 words) - 17:16, 17 September 2012
• 2009 USAMO Problems/Problem 5
  ...> is cyclic if and only if <math>\overline{BG}</math> bisects <math>\angle CBD</math>. pair A = (-.6, .8), B = (.6, .8), C = (.9, -sqrt(.19)), D = (-.9, -sqrt(.19)), G =
  6 KB (973 words) - 18:24, 18 October 2018
• 2002 AMC 12A Problems/Problem 23
  <math>\text{(A)}\ 14 \qquad \text{(B)}\ 21 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 14\sqr pair A, B, C, D, M;
  6 KB (899 words) - 21:04, 10 November 2024
• 2004 AMC 10B Problems/Problem 24
  <math>\text{(A) } \dfrac{9}{8} \qquad \text{(B) } \dfrac{5}{3} \qquad \text{(C) } 2 \qquad pair A, B, C, D, E, I;
  3 KB (513 words) - 14:41, 2 November 2024

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