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• Points $A$, $B$, $C$, and $D$ lie on a line, in that order. If $AB=2$ units, $BC=5$ units and $AD=14$ units, what is the ratio of $AC$ to $BD$? Express your answer as a common fraction.
  <math>AC = AB + BC = 7</math>. <math>BD = AD - AB = 12</math>. Thus, <math>AC:BD=\boxed{\frac{
  139 bytes (21 words) - 20:18, 20 October 2024
• In triangle ABC, D be a point in BC, ∠BAD = 30 , ∠CAD = 90 , BD=1=AC, DC =
  0 bytes (0 words) - 12:26, 6 February 2024

Page text matches

• Pythagorean Theorem
  ...n [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building right tria In these proofs, we will let <math>ABC </math> be any right triangle with a right angle at <math>\angle ACB</math>.
  5 KB (885 words) - 22:03, 5 October 2024
• Math textbooks
  This '''Math textbooks''' page is for compiling a list of [[textbook]]s for mathematics -- not problem books, contest books, ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato
  7 KB (901 words) - 13:11, 6 January 2022
• Physics
  <strong>Physics</strong> is a branch of [[science]] that studies the properties of matter, energy, and ma Modern Physics is also a group of different subjects in physics:
  9 KB (1,355 words) - 06:29, 29 September 2021
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (709 words) - 14:00, 1 June 2024
• Trivial Inequality
  <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
  3 KB (583 words) - 20:20, 2 August 2024
• Heron's Formula
  ...rmula) is a [[mathematical formula | formula]] for finding the [[area]] of a [[triangle]] given only the three side lengths. ...any triangle with side lengths <math>{a}, {b}, {c}</math>, the area <math>{A}</math> can be found using the following formula:
  4 KB (675 words) - 22:45, 6 October 2024
• Factoring
  ...rs, is to write the expression (often an [[integer]] or [[polynomial]]) as a product of different terms. This often allows one to find information abou Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for differe
  3 KB (532 words) - 21:00, 13 January 2024
• 2020 CIME II Problems/Problem 10
  ...n <math>a+b+1=ab\Rightarrow (a-1)(b-1)=2</math> from whence we have <math>(a,b,c)\in\{(2,3,1),(3,2,1)\}</math>. ...math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Rightarrow c=1</math>; a contradiction since <math>c\geq 2</math>.
  2 KB (332 words) - 08:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ..._2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 ...cdot \overrightarrow{w}|</cmath> with equality if and only if there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarro
  13 KB (2,048 words) - 14:28, 22 February 2024
• Menelaus' theorem
  ...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. ...>AB</math> on <math>\triangle ABC</math>, where <math>P</math> is on <math>BC</math>, <math>Q</math> is on the extension of <math>AC</math>, and <math>R<
  5 KB (804 words) - 02:01, 12 June 2023
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,660 words) - 22:42, 28 August 2024
• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ...level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]].
  7 KB (1,300 words) - 00:11, 28 October 2024
• Brahmagupta's Formula
  ...ath>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. ...^2+b^2-2ab\cos B=c^2+d^2-2cd\cos D</math>. <math>\cos B=-\cos D</math>, so a little rearranging gives
  3 KB (543 words) - 18:35, 29 October 2024
• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 23:28, 19 September 2024
• Elementary symmetric sum
  An '''elementary symmetric sum''' is a type of [[summation]]. ...>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  2 KB (275 words) - 11:51, 26 July 2023
• Number theory/Advanced
  ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
  5 KB (849 words) - 15:14, 18 May 2021
• Incenter/excenter lemma
  ...ircumcenter. Then, the theorem states that <math>L</math> is the center of a circle through <math>I</math>, <math>B</math>, <math>I_A</math>, and <math> ...try. Along with the larger lemma, two smaller results follow: first, <math>A</math>, <math>I</math>, <math>L</math>, and <math>I_A</math> are collinear,
  2 KB (291 words) - 15:31, 18 May 2021
• Orthic triangle
  ...h>E</math>, and <math>F</math> denote the feet of the altitudes from <math>A</math>, <math>B</math>, and <math>C</math>, respectively. Then, <math>\tria ...is the orthic triangle of <math>\triangle ABC</math> if and only if <math>A</math>, <math>B</math>, and <math>C</math> are the excenters of <math>\tria
  8 KB (1,408 words) - 08:39, 10 July 2024
• Common factorizations
  ...e be solved by [[Newton sums]] or problems that give a polynomial, and ask a question about the roots. Combined with [[Vieta's formulas]], these are ex \text{\textbullet}&&a^2+b^2+c^2+2(ab+bc+ac)&=(a+b+c)^2\\
  2 KB (327 words) - 01:06, 28 April 2024
• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. The inequality states that in for four points <math>A, B, C, D </math> in the plane,
  3 KB (602 words) - 08:01, 7 June 2023

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