...ath> intersect diagonal <math>BD</math> at points <math>M</math> and <math>N,</math> respectively. It is known that <math>BM = ND, BC \ne CD.</math>
Denote <math>\angle A = 2 \alpha, \angle B = 2 \beta, \angle C = 2 \gamma.</math>
...
122 KB (21,381 words) - 13:47, 3 August 2025
...ath>G</math>, [[circumcenter]] <math>O</math>, [[nine-point center]] <math>N</math> and [[De Longchamps point | de Longchamps point]] <math>L</math>. I
...ath>, and <math>\triangle CH_AH_B</math> [[concurrence | concur]] at <math>N</math>, the nine-point circle of <math>\triangle ABC</math>.
...
59 KB (10,203 words) - 03:47, 30 August 2023
...ircle <math>AA'P, \omega </math> is circle <math>BB'P, x_0 = \Omega \cap \omega, x_0 \neq P,</math>
...ath>BB'), \omega </math> is circle tangent to <math>AB, x_0 = \Omega \cap \omega, x_0 \neq B, C </math> is any point of <math>AB, \theta </math> is circle <
...
28 KB (4,687 words) - 21:27, 19 February 2025
Let a circle <math>\omega</math> with diameter <math>PQ</math> and a point <math>A</math> on this dia
...the point <math>A</math> into point <math>O'</math> - the center of <math>\omega'.</math>
...
43 KB (7,623 words) - 19:07, 24 September 2025
...<math>O_1</math> and <math>O_2</math>, respectively. A third circle <math>\Omega</math> passing through <math>O_1</math> and <math>O_2</math> intersects <ma
label("$B$",B,N+1/2*E);
...
18 KB (2,857 words) - 20:16, 21 September 2025
...gle, <math>a, b, c,</math> be the lengths of <math>BC, AC, AB, \angle A = \alpha, \angle B = \beta, \angle C = \gamma.</math>
<math>\cos \omega = \frac {a^2 + b^2 +c^2}{2 \cdot 2S}</math> is the cosine of the Brocard an
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38 KB (7,569 words) - 02:35, 16 September 2025
...mmutative. For example <math>1+\omega=\omega</math>, while <math>\omega+1>\omega</math>.
...an it. For example, <math>0,1,2,\dotsb,\omega</math> has order type <math>\omega+1</math>.
...
5 KB (811 words) - 13:16, 7 June 2020
.../3}</math>, then we have <math>C=0</math>, <math>B=r</math>, and <math>A=r\omega.</math> Recall that counter-clockwise rotation in the complex plane by an
...a}(D_1-A)</math>, <math>E_3=A+\omega(D_2-A)</math>, <math>E_4=A+\overline{\omega}(D_2-A)</math>. Thus:
...
14 KB (2,311 words) - 19:55, 9 June 2025
...triangle ABC (\angle BAC > \angle BCA)</math> and it’s circumcircle <math>\Omega</math> be given.
The tangent to <math>\Omega</math> at <math>B</math> meet <math>AC</math> at point <math>M.</math>
...
17 KB (3,026 words) - 13:33, 17 March 2025
...e incenter) touch the side <math>BC</math> at point <math>M, \angle BAM = \alpha,</math>
<math>\angle ABC = 2 \beta, \alpha + 2 \beta = 90^\circ, MI = r(</math> inradius).
...
18 KB (3,159 words) - 14:32, 16 September 2025
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
...math>BC</math> lies on <math>\gamma</math>, and <math>BC</math> and <math>\omega</math> are swapped. Thus points <math>F</math> and <math>M</math> map to ea
...
15 KB (2,516 words) - 16:28, 17 September 2024
...</math>, and the other two excircles are both externally tangent to <math>\omega</math>. Find the minimum possible value of the perimeter of <math>\triangle
...th> (using the exradius formula). However, the sum of the radius of <math>\omega</math> and <math>\frac{rs}{s-b}</math> is equivalent to the distance betwee
...
25 KB (4,662 words) - 13:32, 26 March 2025
Let triangle <math>ABC</math> with circumcircle <math>\Omega</math> and points <math>D</math> and <math>E</math> on the sides <math>AB</
Let <math>K \in \Omega</math> be the midpoint of the arc <math>BC</math> which contain the point <
...
10 KB (1,745 words) - 11:28, 21 September 2025
Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
<center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center>
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4 KB (704 words) - 10:25, 28 July 2025
...is the complex conjugate of omega (note that we may assume that <math>\Im(\omega)>0</math>).
...\mathbb{Z}</math> is equivalent to the desired property. Let <math>\omega=\alpha+i\beta</math>.
...
8 KB (1,405 words) - 01:47, 26 November 2024
...AB = 3</math>, <math>BC = 4</math>, and <math>CA = 5</math>. Circle <math>\omega</math> intersects <math>\overline{AB}</math> at <math>E</math> and <math>B<
...th>\angle DBE = 90^\circ</math>, <math>DE</math> is the diameter of <math>\omega</math>. Then <math>\angle DFE=\angle DGE=90^\circ</math>. But <math>DF=FE</
...
17 KB (2,743 words) - 09:25, 7 November 2025
...sed as the quadratic polynomial <cmath>Area(PQRS) = \alpha \omega - \beta \omega^2.</cmath>
...h> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
...
8 KB (1,298 words) - 05:43, 5 February 2025
...rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>?
label("$D$",D,N);
...
20 KB (3,497 words) - 14:43, 14 August 2025
<cmath>\sum_{n=2}^\infty \frac{1}{n^m (\log{n})^p}</cmath>
...f there is a neighborhood <math>N</math> of <math>p</math> such that <math>N \subseteq E</math>
...
13 KB (2,473 words) - 20:53, 10 January 2025
string[] subscripts={"$1$","$2$"," ","$n$","$1$"};
...be written in form of a summation <cmath>A = \dfrac{1}{2} \left|\sum_{i=1}^n{(x_{i+1}+x_i)(y_{i+1}-y_i)}\right|</cmath>
...
8 KB (1,360 words) - 17:09, 28 September 2024
