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• Isogonal conjugate
  Let <math>\theta = \angle ACP = \angle BCQ, \Theta = \angle ACQ = \angle BCP.</math> ...= \frac {PC \sin \theta}{PC \sin \Theta} = \frac {QC \sin \theta}{QC \sin \Theta} = \frac {QD'}{QE'}. \blacksquare</cmath>
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 12A Problems/Problem 20
  label("\(\sin \theta = \frac{3}{5}\)",B-(.2,-.1),W); ...<math>CD</math> tangent to <math>O_a</math> <math>M</math>, and the point tangent to <math>O_b</math> <math>N</math>. Since <math>\triangle CO_aM</math> and
  6 KB (951 words) - 16:31, 2 August 2019
• 2008 AMC 12A Problems/Problem 24
  label("\(\theta\)",(7,.4)); Let <math>x = CA</math>. Then <math>\tan\theta = \tan(\angle BAF - \angle DAE)</math>, and since <math>\tan\angle BAF = \f
  3 KB (513 words) - 14:35, 7 June 2018
• 2008 AIME I Problems/Problem 14
  ...<math>T</math> lies on <math>\omega</math> so that line <math>CT</math> is tangent to <math>\omega</math>. Point <math>P</math> is the foot of the perpendicul label("\(\theta\)",C + (-1.7,-0.2), NW);
  8 KB (1,333 words) - 00:18, 1 February 2024
• 2008 AIME II Problems/Problem 11
  .... Circle <math>Q</math> is externally [[tangent]] to <math>P</math> and is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math>. No point of c ...t <math>\angle ACB = 2\theta</math>; then <math>\angle PCX = \angle QBX = \theta</math>. Dropping the altitude from <math>A</math> to <math>BC</math>, we re
  6 KB (1,065 words) - 20:12, 9 August 2022
• 1998 AHSME Problems/Problem 28
  ...nd <math>3 \theta = \angle CAB</math>. Then, it is given that <math>\cos 2\theta = \frac{AC}{AD} = \frac{2}{3}</math> and ...tan 3\theta - \tan 2\theta)}{AC \tan 2\theta} = \frac{\tan 3\theta}{\tan 2\theta} - 1.</math></center>
  4 KB (662 words) - 00:51, 3 October 2023
• 2008 iTest Problems
  ...sible values of <math>\tan\theta</math> given that <math>\sin\theta + \cos\theta = \dfrac{193}{137}</math>. If <math>a+b=m/n</math>, where <math>m</math> an ...ists a circle, lying inside the quadrilateral and having center I, that is tangent to all four sides of the quadrilateral. Points <math>M</math> and <math>N</
  71 KB (11,749 words) - 01:31, 2 November 2023
• Spiral similarity
  ...ath> is any point of <math>AB, \theta </math> is circle <math>CPx_0, C' = \theta \cap A'B'</math> is the image <math>C</math> under spiral symilarity center ...x_0 = \Omega \cap \omega, x_0 \neq B, C </math> is any point of <math>AB, \theta </math> is circle <math>CBx_0,</math>
  28 KB (4,863 words) - 00:29, 16 December 2023
• 1992 USAMO Problems/Problem 4
  ...ough <math>A</math>, <math>B</math>, <math>C</math>, and <math>P</math> is tangent to the sphere through <math>A'</math>, <math>B'</math>, <math>C'</math>, an ...e that pass through <math>P</math> for the <math>2</math> spheres that are tangent to each other.
  5 KB (807 words) - 18:37, 25 June 2021
• 2009 AMC 10B Problems/Problem 20
  ...\angle DAC</math>. Notice <math>\tan \theta = BD</math> and <math>\tan 2 \theta = 2</math>. By the double angle identity, <cmath>2 = \frac{2 BD}{1 - BD^2} Remarks: You could also use tangent half angle formula
  2 KB (371 words) - 15:34, 15 October 2023
• 2009 AIME I Problems/Problem 12
  ...ch that <math>\overline{AI}</math> and <math>\overline{BI}</math> are both tangent to circle <math>\omega</math>. The ratio of the perimeter of <math>\triangl ...}}</math>. By the double-angle formula <math>\sin(2\theta)=2\sin\theta\cos\theta</math>, it turns out that <cmath>\sin(\angle BAI)=\sin(2\angle DAO)=\dfrac{
  12 KB (1,970 words) - 22:53, 22 January 2024
• 2010 AMC 12B Problems/Problem 18
  ...le, which is itself centered on <math>(1,0)</math>. Let us define <math>f(\theta)</math> as the value of the length of the first circle that lies within the <cmath>f(\theta)=\frac{\theta}{2\pi}\cdot2\pi=\theta</cmath>
  6 KB (1,105 words) - 13:39, 9 January 2024
• 2011 AMC 12A Problems/Problem 24
  ...ath>\angle ABC=180^{\circ}-\alpha</math> and <math>\angle BCD=180^{\circ}-\theta</math>. Let the circle have center <math>O</math> and radius <math>r</math> ...\theta</math>, <math>GOH=180^{\circ}-\alpha</math>, <math>EOH=180^{\circ}-\theta</math>, and <math>FOE=\alpha</math>.
  4 KB (753 words) - 18:58, 2 June 2022
• 2011 USAJMO Problems/Problem 5
  ...ven points are such that (i) lines <math>PB</math> and <math>PD</math> are tangent to <math>\omega</math>, (ii) <math>P</math>, <math>A</math>, <math>C</math> Since PB and PD are tangent to the circle, it's easy to see that M is the midpoint of arc BD.
  4 KB (717 words) - 17:00, 14 April 2024
• 2011 USAJMO Problems/Problem 3
  ...nnot be on <math>p</math>. This implies that <math>q</math> is exactly the tangent line to <math>p</math> at <math>P</math>, that is <math>q=\ell(P)</math>. S ...h> be the reflection of <math>F</math> across <math>q</math>. Then <math>2\theta=\angle FBH=\angle C'HB</math>, and so <math>\angle C'HB=\angle AHB</math>.
  15 KB (2,593 words) - 13:37, 29 January 2021
• 1984 AHSME Problems/Problem 29
  ...math>OA=\sqrt{18}</math>, <math>\angle OXA = 90</math> because the line is tangent to the circle. Using the pythagorean theorem, we have <math>OX=\sqrt{12}</m ...th>\tan (\theta + 45)</math> where <math>\angle AOX = \theta</math>. Using tangent addition,
  4 KB (614 words) - 20:09, 12 September 2022
• 1979 USAMO Problems/Problem 2
  ...the equator, then <math>C=(cos(\theta),sin(\theta),0)</math> where <math>\theta</math> is the angle on the <math>xy</math>-plane from the origin to <math>C ...CN}}</math> is the unit vector in the direction of arc <math>CN</math> and tangent to the great circle of <math>CN</math> at <math>C</math>
  6 KB (1,013 words) - 22:09, 21 November 2023
• 1996 AHSME Problems/Problem 25
  ...misses the circle altogether. This means <math>3x + 4y = A</math> will be tangent to the circle. ...will be <math>\frac{4}{3}</math>, since the radius is perpendicular to the tangent line.
  9 KB (1,441 words) - 17:51, 22 October 2023
• Barycentric coordinates
  The incircle contains the tangent points of the incircle with the sides: Denote <math>R = PQ \cap P'Q', \theta = \odot P'QR, F = \Omega \cap \theta \notin \odot PQ'R, D \in \Omega</math> is the point isogonal conjugate to l
  25 KB (5,067 words) - 22:15, 31 March 2024
• Mathtime Version 1 Issue 2
  ...e has a circle inscribed within the circle so that the inscribed circle is tangent to all <math>3</math> sides. What is the ratio of the sum of the areas of a ...ouches the origin and has its center located on the y-axis. The circle is tangent to the line <math>x + 3y = 6</math>. Given that the radius of the circle c
  15 KB (2,444 words) - 21:46, 1 January 2012

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