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• 2005 AIME I Problems/Problem 15
  Triangle <math> ABC </math> has <math> BC=20. </math> The [[incircle]] of the triangle evenly [ Now, by [[Stewart's Theorem]] in triangle <math>\triangle ABC</math> with [[cevian]] <math>\overline{AD}</math>, we have
  5 KB (906 words) - 22:15, 6 January 2024
• 1984 AIME Problems/Problem 9
  ...ath>, [[edge]] <math>AB</math> has length 3 cm. The area of [[face]] <math>ABC</math> is <math>15\mbox{cm}^2</math> and the area of face <math>ABD</math> ...= 8</math>. Because the problem does not specify, we may assume both <math>ABC</math> and <math>ABD</math> to be isosceles triangles. Thus, the height of
  6 KB (947 words) - 19:44, 26 November 2021
• 1989 AIME Problems/Problem 10
  Let <math>K</math> be the area of <math>\triangle ABC.</math> Then <math>h=\frac{2K}{c}</math>, so <math>\cot{\alpha}+\cot{\beta} ...}{\sin \gamma} = \frac{a^2 + b^2 - c^2}{2ab} \cdot \frac{2R}{c} = \frac{R}{abc} (a^2 + b^2 - c^2). </cmath>
  8 KB (1,401 words) - 20:41, 20 January 2024
• 1998 AIME Problems/Problem 12
  ...n <math>\overline{BP}.</math> The [[ratio]] of the area of triangle <math>ABC</math> to the area of triangle <math>PQR</math> is <math>a + b\sqrt {c},</m WLOG, assume that <math>AB = BC = AC = 2</math>.
  5 KB (884 words) - 13:33, 18 June 2024
• 2001 AIME I Problems/Problem 5
  ...endpoints of the base are <math>(x,y)</math> and <math>(-x,y)</math>, and WLOG, we can say that <math>x</math> is positive. ...lengths <math>2n, n\sqrt{13}, n\sqrt{13}.</math> Let the triangle be <math>ABC</math> such that <math>AB=AC.</math> Let the foot of the altitude from A be
  6 KB (1,043 words) - 09:09, 15 January 2024
• 2003 AIME I Problems/Problem 10
  [[Triangle]] <math> ABC </math> is [[isosceles triangle | isosceles]] with <math> AC = BC </math> a /* We will WLOG AB = 2 to draw following */
  7 KB (1,082 words) - 01:08, 30 September 2024
• 2002 AIME II Problems/Problem 13
  ...tio of the area of triangle <math>PQR</math> to the area of triangle <math>ABC</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are rel ...ta ABC \sim \Delta QRP</math>, and thus, <math>\frac{[\Delta PQR]}{[\Delta ABC]} = \left(\frac{PX}{CX}\right)^2</math>.
  6 KB (937 words) - 19:06, 24 August 2024
• Euler line
  In any [[triangle]] <math>\triangle ABC</math>, the '''Euler line''' is a [[line]] which passes through the [[ortho ...nce | concur]] at <math>N</math>, the nine-point circle of <math>\triangle ABC</math>.
  59 KB (10,203 words) - 03:47, 30 August 2023
• Fermat point
  ...point''' (also called the Torricelli point) of a triangle <math>\triangle ABC</math> (with no angle more than <math>120^{\circ}</math> is a point <math>P ...ct three equilateral triangles out of the three sides from <math>\triangle ABC</math>, then connect each new vertex to each opposite vertex, as these thre
  4 KB (769 words) - 15:07, 29 December 2019
• Concurrence/Problems
  ...respectively). Warren gets a little confused and draws a certain triangle ABC along with the median from vertex A, the altitude from vertex B, and the an WLOG, assuming that the line segment is <math>AE</math>:
  3 KB (458 words) - 14:44, 1 December 2015
• De Longchamps point
  We call A-power circle of a <math>\triangle ABC</math> the circle centered at the midpoint <math>BC</math> point <math>A'</ <math>a = BC, b = AC, c = AB.</math> WLOG, <math>a \ge b \ge c.</math>
  10 KB (1,782 words) - 19:35, 28 September 2024
• 1995 AHSME Problems/Problem 29
  We can account for permutations by assuming WLOG that <math>a</math> contains the prime factor 2. Thus, there are <math>3^4< ...3 \cdot 5 \cdot 7 \cdot 11.</math> Therefore, we have the equation <cmath> abc = 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11,</cmath>where <math>a, b, c</ma
  2 KB (371 words) - 13:33, 15 January 2023
• Isogonal conjugate
  Let triangle <math>ABC</math> be given. Let <math>AD</math> and <math>AE</math> be the isogonals w ...AB</math> and <math>AC</math> to the outer (inner) side of <math>\triangle ABC.</math> Let <math>\angle BAD = \angle CAE, H = CD \cap BE.</math>
  54 KB (9,416 words) - 07:40, 18 April 2024
• Mass points
  ...irst choosing a point for the entire figure to balance around. From there, WLOG a first weight can be assigned. From the first weight, others can be derive ...math> and <math>AC</math> respectively. Let the centroid of triangle <math>ABC</math> be <math>P</math>. Prove <math>DP:PC</math> is <math>1:2</math> and
  5 KB (849 words) - 13:27, 8 August 2024
• 2003 USAMO Problems/Problem 5
  Since all terms are homogeneous, we may assume WLOG that <math>a + b + c = 3</math>. <cmath>\sum_{\text{sym}} a^3 - 2a^2b + abc\geq 0.</cmath>
  6 KB (1,063 words) - 01:36, 9 August 2023
• 1972 USAMO Problems/Problem 2
  ...at are perpendicular to side <math>AB</math>. We have that triangles <math>ABC</math> and <math>ABD</math> are non-acute, so <math>C</math> and <math>D</m ...cdot\vec{c}</math>, and <math>\vec{c}\cdot\vec{a}</math> are all positive. WLOG, <math>\vec{a}\cdot\vec{a}\geq \vec{b}\cdot\vec{b}, \vec{c}\cdot\vec{c} > 0
  8 KB (1,458 words) - 22:42, 27 February 2022
• 1974 USAMO Problems/Problem 2
  <center><math>a^ab^bc^c\ge (abc)^{(a+b+c)/3}</math></center> <center><math>\frac{a+b+c}{3}\ge \sqrt[3]{abc}</math></center>
  9 KB (1,744 words) - 09:20, 24 August 2024
• 2004 AMC 10B Problems/Problem 20
  In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>BC</math> and WLOG, we can assume <math>[\triangle BTD] = 1</math>. Then <math>[\triangle BTA]
  6 KB (950 words) - 13:35, 13 October 2024
• 2009 AMC 10A Problems/Problem 24
  ...th> vertices is the third vertex <math>C</math>, the entire triangle <math>ABC</math> will be on a face. On the other hand, if <math>C</math> is one of th ...on the side that contains <math>AB</math>. Hence the probability of <math>ABC</math> intersecting the inside of the cube is <math>2/3</math>.
  5 KB (795 words) - 19:25, 31 October 2022
• 2007 AMC 12B Problems/Problem 22
  ...is the ratio of the area of <math>R</math> to the area of <math>\triangle ABC</math>? [[WLOG]], let point <math>A</math> be located on the origin. Let side <math>\overl
  4 KB (707 words) - 15:36, 15 February 2021

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