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• Inverse with respect to an operation
  30 bytes (3 words) - 11:39, 23 November 2007

Page text matches

• Pythagorean Theorem
  ...considering the [[Power of a Point | power of point]] <math>A </math> with respect to <math>\omega </math>, we see
  5 KB (886 words) - 21:12, 22 January 2024
• Physics
  * The rate of change of momentum of a body with respect to time is directly proportional to the force acting on it.
  9 KB (1,355 words) - 07:29, 29 September 2021
• Cyclic quadrilateral
  * Any two opposite sites of the quadrilateral are antiparallel with respect to the other two opposite sites.
  1 KB (162 words) - 20:39, 9 March 2024
• Riemann sum
  The '''Riemann sum''' of <math>f</math> with respect to <math>\mathcal{\dot{P}}</math> on <math>[a,b]</math> is defined as <math The '''Upper sum''' of <math>f</math> with respect to <math>\mathcal{P}</math> on <math>[a,b]</math> is defined as <math>U(f,\
  1 KB (272 words) - 12:06, 7 May 2008
• Henstock-Kurzweil integral
  .../math> is the [[Riemann sum]] of <math>f</math> on <math>[a,b]</math> with respect to <math>\mathcal{\dot{P}}</math>
  2 KB (401 words) - 09:46, 31 January 2018
• Pi
  Integrating <math>n</math> times by parts with respect to <math>x</math>, we see that our integral equals <math>\iint_{(0,1)\times
  8 KB (1,469 words) - 21:11, 16 September 2022
• Homothety
  ...'') is a transformation of space which [[dilation|dilates]] distances with respect to a fixed point. A homothety can be an enlargement (resulting figure is l ...itionally, <math>ABCD</math> and <math>A'B'C'D'</math> are homothetic with respect to <math>O</math>.
  3 KB (532 words) - 01:11, 11 January 2021
• Trigonometry
  ...f trigonometry involves the ratios between edges of a right triangle, with respect to a given angle. The definitions below will be referring to angle <math>A<
  8 KB (1,217 words) - 20:15, 7 September 2023
• Power of a Point Theorem
  .../math> must be a circle. If we consider the lines and <math>P</math> with respect simply to that circle, then we have reduced our claim to the case of two di
  5 KB (827 words) - 17:30, 21 February 2024
• Mathematicial notation
  is the "Hilbert symbol of <math>a,b</math> in respect to <math>v</math>" (nontrivial means here that not all numbers are <math>0<
  8 KB (1,401 words) - 13:11, 17 June 2008
• Algebraic geometry
  ...are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology.
  2 KB (361 words) - 01:59, 24 January 2020
• Algebraic topology
  ...a(t)</math>, which is "compatible" with the concatenation in the following respect:
  3 KB (479 words) - 15:35, 1 December 2015
• 2005 AMC 12B Problems/Problem 7
  The graph is symmetric with respect to both coordinate axes, and in the first quadrant it coincides with the gr
  2 KB (357 words) - 20:15, 27 December 2020
• 2005 AIME II Problems/Problem 14
  ==Solution 5 (Isogonal lines with respect to A angle bisesector)== Since <math>AE</math> and <math>AD</math> are isogonal with respect to the <math>A</math> angle bisector, we have <cmath>\frac{BE}{EC}\cdot \fr
  13 KB (2,129 words) - 18:56, 1 January 2024
• 1983 AIME Problems/Problem 11
  ...th>l=6\sqrt{2}+\sqrt{2}h</math> since it similarly increases linearly with respect to <math>h</math>. Now we solve:<cmath>\int_0^6(6\sqrt{2}-\sqrt{2}h)(6\sqrt
  5 KB (865 words) - 21:11, 6 February 2023
• 1984 AIME Problems/Problem 15
  ...}{a-5^2}+\frac{w^2}{a-7^2}=1</math>. We can let this be a polynomial, with respect to <math>a</math>. We can say that <math>w^2</math>, <math>x^2</math>, <mat
  6 KB (1,050 words) - 18:07, 16 January 2024
• 1986 AIME Problems/Problem 2
  ...this solution, note that the original expression has cyclic symmetry with respect to <math>x,y,</math> and <math>z:</math>
  3 KB (460 words) - 00:44, 5 February 2022
• 1996 AIME Problems/Problem 7
  ...re isn't occupied and the two squares that are opposite to each other with respect to the center square, in which there are <math>12</math> cases. ...ach other with a <math>90^{\circ}</math> rotation with each other and with respect to the center square, in which case there are <math>12</math> cases.
  4 KB (551 words) - 11:44, 26 June 2020
• 1997 AIME Problems/Problem 12
  ...ath> and <math>97</math>, we know that the involution is an inversion with respect to a circle with a diameter from <math>19</math> to <math>97</math>. The on
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1998 AIME Problems/Problem 9
  ...\frac{2m}{60}</math>, depending upon the location of <math>M_1</math> with respect to the endpoints. Intuitively, the average probability will occur at <math>
  4 KB (624 words) - 18:34, 18 February 2018
• 2000 AIME I Problems/Problem 13
  ...at <math>(x,0)</math>. All these circles are [[homothety|homothetic]] with respect to a center at <math>(5,0)</math>.
  3 KB (571 words) - 00:38, 13 March 2014
• 2001 AIME I Problems/Problem 5
  Since the triangle is symmetric with respect to the y-axis, the coordinates of <math>B</math> and <math>C</math> are now
  6 KB (1,043 words) - 10:09, 15 January 2024
• 2003 AIME I Problems/Problem 11
  Note that the three expressions are symmetric with respect to interchanging <math>\sin</math> and <math>\cos</math>, and so the probab
  2 KB (284 words) - 13:42, 10 October 2020
• Negative number
  * [[Inverse with respect to an operation]]
  635 bytes (93 words) - 14:07, 16 January 2023
• Centroid
  ...spectively. Hence the triangles <math>ABC, DEF</math> are homothetic with respect to some point <math>G</math> with dilation factor <math>-\frac{1}{2}</math>
  3 KB (612 words) - 22:32, 25 February 2024
• Group
  ...>g^{-1}\in G</math> so that <math>gg^{-1}=g^{-1}g=e</math> ([[Inverse with respect to an operation | inverses]]).
  2 KB (365 words) - 12:03, 12 November 2023
• Inverse
  * [[Inverse with respect to an operation]], such as in a [[group]] (see also [[identity]])
  334 bytes (52 words) - 11:42, 23 November 2007
• Holomorphic function
  ...all the [[partial derivative]]s of <math>u</math> and <math>v</math> with respect
  9 KB (1,537 words) - 21:04, 26 July 2017
• Dedekind domain
  ...t <math>M_f</math> be the [[matrix]] representation of <math>f</math> with respect to <math>x_1,\ldots,x_m</math>. Then <math>M_f</math> is an <math>m\times m
  9 KB (1,648 words) - 16:36, 14 October 2017
• Operation
  * [[Inverse with respect to an operation]]
  357 bytes (54 words) - 09:17, 18 July 2006
• Vector space
  ...ity]] (usually denoted <math>\mathbf 0</math>) and additive [[inverse with respect to an operation | inverses]].
  3 KB (561 words) - 00:47, 21 March 2009
• Mock AIME 1 2006-2007 Problems/Problem 5
  ...to the following: "If <math>n</math> has a multiplicative [[inverse with respect to an operation | inverse]] [[mod]] <math>p</math>, <math>f_p(n)</math> is
  2 KB (340 words) - 15:52, 3 April 2012
• Mock AIME 1 2006-2007 Problems/Problem 7
  ...<math>BF</math> is 1. We can now use Menelaus on line <math>DF</math> with respect to triangle <math>ABC</math>:
  3 KB (518 words) - 16:54, 25 November 2015
• 1973 IMO Shortlist Problems/Bulgaria 1
  ...nce the [[power of a point | power]] of <math>\displaystyle P </math> with respect to <math>\displaystyle S </math> is invariant, we may multiply both sides o
  2 KB (346 words) - 14:59, 30 July 2006
• Derivative
  ...]] is defined as the instantaneous [[rate]] of change of the function with respect to one of the [[variable]]s. Note that not every function has a derivative
  509 bytes (64 words) - 19:26, 3 March 2010
• Reciprocal
  ...any non-zero element of a [[field]]) is its multiplicative [[inverse with respect to an operation | inverse]]. The reciprocal is usually denoted <math>r^{-1
  554 bytes (87 words) - 11:43, 23 November 2007
• Integral
  ...(f,\mathcal{\dot{P}})</math> is the [[Riemann sum]] of <math>f</math> with respect to <math>\mathcal{\dot{P}}</math> ...emann sum|lower sum]] and [[Riemann sum|upper sum]] of <math>f</math> with respect to [[partition]] <math>\mathcal{P}</math>
  5 KB (909 words) - 14:16, 31 May 2022
• Free group
  The [[inverse with respect to an operation | inverse]] of a given element of a free group can be found
  2 KB (454 words) - 17:54, 16 March 2012
• Monoid
  ...natively, a monoid can be thought of as a [[group]] without [[inverse with respect to an operation | inverses]], or as an associative [[magma]] with an identi ...ebra]], and many objects (such as all groups, as well as any [[ring]] with respect to either of its operations) are in fact monoids.
  3 KB (670 words) - 22:45, 21 May 2008
• Euler line
  Let <math>\omega'</math> be the circle symmetric to <math>\omega</math> with respect to <math>AB.</math> Let <math>E</math> be the point symmetric to <math>O</math> with respect to <math>AB.</math>
  59 KB (10,203 words) - 04:47, 30 August 2023
• Special relativity
  ...t moving in the direction of motion of earth should have moved slower with respect to the earth than light moving in the opposite direction, as everything had
  3 KB (538 words) - 23:39, 2 March 2008
• Skew field
  ...ommutative) ring in which every [[element]] has a two-sided [[inverse with respect to an operation | inverse]]. Equivalently, a skew field is a [[field]] in
  1 KB (253 words) - 19:05, 9 September 2008
• Quaternion
  ...t is, a [[ring]] in which each element has a multiplicative [[inverse with respect to an operation | inverse]]; alternatively, a non[[commutative]] [[field]])
  1,007 bytes (155 words) - 20:47, 14 October 2013
• Canada/USA Mathcamp
  ...thcamp officially has only four rules (generally invoking common sense and respect). Students have the freedom and flexibility to choose what classes and act
  4 KB (680 words) - 23:22, 12 April 2021
• Totally ordered set
  ...natural" ordering of this set. In particular, it behaves very poorly with respect to arithmetic operations on <math>\mathbb C</math>.
  1 KB (174 words) - 12:52, 25 November 2007
• 2005 IMO Shortlist Problems/G1
  ...of <math> \displaystyle D </math> and <math> \displaystyle E </math> with respect to <math> \displaystyle I </math>. Prove that the quadrilateral <math> \di ...yle T </math> to the circumcircle of <math> \displaystyle ABC </math> with respect to lines <math> \displaystyle AC, PT </math> gives us <center> <math> \frac
  7 KB (1,088 words) - 16:57, 30 May 2007
• 2004 IMO Shortlist Problems/G2
  By considering the power of the point <math> \displaystyle D </math> with respect to <math> \displaystyle \Gamma </math>, we see <math> \displaystyle DE^2 =
  10 KB (1,539 words) - 23:37, 6 June 2007
• Circumradius
  Circumradius and altitude are isogonals with respect bisector and vertex of triangle.
  4 KB (729 words) - 16:52, 19 February 2024
• Chicken McNugget Theorem
  ...every integer point <math>T</math>, the reflection of <math>T</math> with respect to <math>P</math> is an integer point of a different colour than <math>T</m
  17 KB (2,748 words) - 19:22, 24 February 2024
• Implicit differentiation
  ...[[derivative|differentiating]] both sides of an implicit [[equation]] with respect to one of the [[variable]]s. The [[dependent variable]] is treated as a [[f
  757 bytes (125 words) - 01:04, 26 March 2018
• 2007 AMC 12A Problems/Problem 22
  ...designed to be satisfied except whether <math>S(n)</math> is accurate with respect to <math>n.</math> So, the only thing that remains is to check this. We may
  15 KB (2,558 words) - 19:33, 4 February 2024
• 2006 AIME II Problems/Problem 12
  ..., and <math>EF=13</math>. If we take a homothety of the parallelogram with respect to <math>A</math>, such that <math>F</math> maps to <math>G</math>, we see
  6 KB (1,033 words) - 02:36, 19 March 2022
• Asymptote: CSE5
  ''S'' allows you to position the label manually with respect to the middle point of the largest arc. The automatic positioning is where
  4 KB (607 words) - 15:09, 30 June 2020
• De Longchamps point
  ...ymmetrical to the point of intersection of the heights <math>H</math> with respect to <math>O.</math> ...<math>C-</math>power cicles is symmetric to altitude <math>AH</math> with respect <math>O.</math>
  10 KB (1,780 words) - 09:23, 17 November 2022
• Construction
  16. Construct the inverse of a point P with respect to circle C. In other words, construct the unique point <math>P'</math> on
  3 KB (443 words) - 20:52, 28 August 2014
• 1990 USAMO Problems/Problem 5
  Now we use the Power of a Point theorem with respect to point <math>H</math>. From the circle with diameter <math>AB</math> we g
  3 KB (604 words) - 20:52, 24 October 2018
• Isogonal conjugate
  '''Isogonal conjugates''' are pairs of [[point]]s in the [[plane]] with respect to a certain [[triangle]]. ...l</math> and containing the point <math>O</math> be called isogonals with respect to the pair <math>(\ell,O).</math>
  54 KB (9,416 words) - 08:40, 18 April 2024
• 2008 AMC 12A Problems/Problem 25
  ...the origin by <math>30^\circ</math> and dilating the point's position with respect to the origin by a factor of <math>2</math>.
  5 KB (745 words) - 10:58, 9 December 2022
• Symmetric group
  ...ect to subgroups, just as [[free group]]s can be considered universal with respect to [[quotient group]]s.
  10 KB (1,668 words) - 15:33, 25 May 2008
• Mock AIME 1 2007-2008 Problems/Problem 14
  The [[Law of Cosines]] on <math>\triangle BAO</math> with respect to <math>\angle BAO</math> yields ...sqrt{11}</math>. The [[Law of Cosines]] on <math>\triangle CAO</math> with respect to <math>\angle CAO</math> yields
  2 KB (375 words) - 15:23, 25 November 2009
• 2008 AIME II Problems/Problem 5
  ...then <math>BC</math> and <math>AD</math> are [[homothecy|homothetic]] with respect to point <math>E</math> by a ratio of <math>\frac{BC}{AD} = \frac{125}{251}
  8 KB (1,338 words) - 23:15, 28 November 2023
• Complement
  ...subset; in this case, we speak of the ''complement of <math>X</math> with respect to <math>A</math>''. Such a set is sometimes denoted <math>\overline{X}</m
  748 bytes (135 words) - 14:22, 9 April 2008
• 2008 AIME II Problems/Problem 13
  ...'ll see that the problem's basically asking you to invert the hexagon with respect to the unit circle in the Cartesian Plane using the Inversion Distance Form
  6 KB (894 words) - 18:56, 25 December 2022
• Fiber product
  ...ategory. Then the fiber product of <math>X</math> and <math>Y</math> with respect to <math>Z</math>, denoted <math>X \times_Z Y</math> (when the specific fun
  770 bytes (131 words) - 21:55, 7 September 2008
• Quotient set
  ...lence relation. The set of [[equivalence class]]es of <math>S</math> with respect to <math>\mathcal{R}</math> is called the ''quotient of <math>S</math> by < ...pect to <math>\mathcal{R}</math>. The ''saturation of <math>A</math> with respect to <math>\mathcal{R}</math>'' is the least saturated subset <math>A'</math>
  2 KB (358 words) - 09:42, 7 June 2016
• Coset
  ...ft cosets of <math>H</math> is called the ''index'' of <math>H</math> with respect to <math>G</math>; it is denoted <math>(G:H)</math>. This is also the card
  4 KB (897 words) - 01:28, 3 September 2008
• Restricted sum
  ...ndices <math>i</math> is the ''restricted sum of the <math>G_i</math> with respect to the <math>H_i</math>''. ...[trivial group |trivial]], the restricted sum of the <math>G_i</math> with respect to the <math>H_i</math> is again the direct sum of the <math>G_i</math>. W
  1 KB (273 words) - 13:07, 6 June 2008
• Mechanics
  *The rate of change of momentum of a body with respect to time is directly proportional to the force acting on it.
  2 KB (309 words) - 15:41, 1 December 2015
• 2003 USAMO Problems/Problem 4
  ...math>. This exact condition can also be visualized through power of M with respect to the circumcircle of triangle <math>BDC</math> and getting the angle cond
  5 KB (820 words) - 02:39, 10 January 2023
• 1995 IMO Problems/Problem 1
  ...XY</math> of the original two circles, so the power of <math>Z</math> with respect to each of the four circles is all equal to <math>ZM * ZA</math>. Hence, <m
  5 KB (847 words) - 19:03, 12 October 2021
• Spiral similarity
  ....</math> The line <math>\ell</math> is symmetrical to <math>SC</math> with respect to <math>AB</math> and intersects <math>BC</math> at <math>D.</math> Prove Points <math>D</math> and <math>D'</math> are simmetric with respect <math>AB,</math> so <math>BD = BD' \implies k = \frac {SC}{SB} = \frac {BC}
  28 KB (4,863 words) - 00:29, 16 December 2023
• 1975 USAMO Problems/Problem 2
  ...math> is a function of <math>\theta</math>, so we take the derivative with respect to <math>\theta</math> and obtain that <math>a^2+b^2</math> takes a minimum ...spond to the position vectors of points A, B, C, and D, respectively, with respect to an arbitrary origin O. Let us also for simplicity define <math>a^2 = a \
  6 KB (1,096 words) - 23:07, 26 August 2017
• 1989 AJHSME Problems/Problem 11
  ...h of the five "T-like shapes" would be [[symmetric]] to the one shown with respect to the dashed line?
  2 KB (192 words) - 22:27, 9 November 2015
• Polynomial ring
  This operator <math>\partial</math> is called the ''(formal) derivative (with respect to <math>x</math>)'', and behaves much like the [[derivative]] of a functio
  12 KB (2,010 words) - 00:10, 3 August 2020
• 1989 AJHSME Problems
  Which of the five "T-like shapes" would be symmetric to the one shown with respect to the dashed line?
  13 KB (1,765 words) - 11:55, 22 November 2023
• Radical axis
  The '''power''' of point <math>P</math> with respect to circle <math>\omega</math> (with radius <math>r</math> and center <math> ...h>P</math> such that the [[power of a point|power]] of <math>P</math> with respect to <math>\omega_1</math> and <math>\omega_2</math> are equal. In other word
  10 KB (1,797 words) - 02:05, 24 October 2023
• 2009 AIME II Problems/Problem 15
  ...1+\tan (\alpha)} - 1</math>. Taking the derivative of <math>VW</math> with respect to <math>\alpha</math>, we arrive at ...math>t</math> (and the constant <math>a</math>). Differentiating this with respect to <math>t</math> yields
  11 KB (1,849 words) - 19:43, 2 January 2023
• 2009 USAMO Problems/Problem 5
  ...= BD \cap TQ</math>, <math>R</math> and <math>R'</math> are symmetric with respect to the axis of symmetry of trapezoid <math>ABCD</math>. This implies that l
  6 KB (973 words) - 19:24, 18 October 2018
• Fundamental group
  ...a(t)</math>, which is "compatible" with the concatenation in the following respect:
  8 KB (1,518 words) - 20:11, 23 January 2017
• Euclidean space
  ...ematical objects. For example, Euclidean space is a [[metric space]] with respect to the [[distance formula|distance]] [[metric]], <math>d(\bold{x},\bold{y})
  1,018 bytes (164 words) - 12:46, 13 March 2010
• Linearly independent
  Any [[eigenvector]]s corresponding to different [[eigenvalue]]s (with respect to a linear map <math>L</math>) are linearly independent. This can be prove
  2 KB (300 words) - 23:35, 16 March 2010
• AXA Achievement Scholarship
  * Respect for self, family and community
  818 bytes (107 words) - 16:12, 14 June 2010
• 2010 IMO Problems/Problem 4
  Now consider the power of point <math>S</math> with respect to <math>\Gamma</math>.
  6 KB (895 words) - 18:32, 2 April 2021
• 2019 IMO Problems/Problem 6
  An inversion with respect <math>\omega</math> swap <math>A</math> and <math>A' \implies A'</math> is ...ga</math> meet on the polar line of the intersection of the diagonals with respect to <math>\omega \implies DI</math> and <math>PS</math> meet on the line thr
  5 KB (792 words) - 01:52, 19 November 2023
• Mobius function
  Let <math>f(n)</math> be a multiplicative function with respect to <math>n</math>. If there is such an <math>f</math>, then
  5 KB (910 words) - 02:58, 1 March 2022
• 2011 AMC 12A Problems
  ...peed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock <math>B</math> downriver, then im
  13 KB (1,994 words) - 13:52, 3 July 2021
• 2011 AMC 12A Problems/Problem 12
  ...peed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock <math>B</math> downriver, then im .../math> distance apart in the third diagram when they meet. Therefore, with respect to each other, the boat and the raft cover a distance of <math>d</math> on
  5 KB (928 words) - 23:01, 14 April 2022
• 2011 AIME II Problems/Problem 10
  ...<math>PB</math> and <math>PA</math>). Then, by Power of a Point on P with respect to the circle with center <math>O</math>, we have that
  11 KB (1,720 words) - 03:12, 18 December 2023
• 2011 USAMO Problems/Problem 5
  ...h>AD</math> or <math>BC</math> depending on where <math>R</math> is). With respect to triangle <math>BCR</math>, <math>P</math> and <math>Q_1</math> are isogo
  4 KB (660 words) - 01:04, 15 February 2024
• 2015 IMO Problems/Problem 4
  ...nd and the third chords are anti-parallel to the first (common) chord with respect to the given lines, so they are parallel to each other. <math>\Box</math><b
  3 KB (502 words) - 23:58, 5 October 2015
• 1996 AHSME Problems/Problem 25
  Implicitly differentiating the given equation with respect to <math>x</math> yields:
  9 KB (1,441 words) - 17:51, 22 October 2023
• 1983 USAMO Problems/Problem 1
  ...and <math>F</math> are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles <math>ABC</
  2 KB (297 words) - 22:29, 17 July 2016
• Barycentric coordinates
  The point <math>P = (x : y : z)</math> is isotomically conjugate with respect to <math>\triangle ABC</math> with the point <math>P_1 =\left( \frac {1}{x} The point <math>P = (x : y : z)</math> is isogonally conjugate with respect to <math>\triangle ABC</math> with the point <math>P_2
  25 KB (5,067 words) - 22:15, 31 March 2024
• 1989 AHSME Problems
  ...in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red
  15 KB (2,247 words) - 13:44, 19 February 2020
• 2012 AMC 12A Problems/Problem 12
  Like the previous solutions, we can say that the square is symmetrical with respect to the y-axis.
  3 KB (574 words) - 20:42, 3 January 2020
• 2012 AMC 12A Problems/Problem 16
  ...th> and <math>XA=6</math>. Consider the power of point <math>Z</math> with respect to Circle <math>O,</math> we have <math>13\cdot 7 = (11 + r)(11 - r) = 11^2
  9 KB (1,496 words) - 02:40, 2 October 2022
• 2012 AIME I Problems/Problem 14
  ...y real number. WLOG, assume that this is the right angle. With symmetry to respect of the x axis (because symmetry makes the imaginary parts of the other 2 ro
  12 KB (2,228 words) - 23:59, 20 January 2024
• 2012 AIME II Problems/Problem 15
  We consider the inversion with respect <math>\theta.</math> ...B' = AC, B' \in AB \implies B'</math> is symmetric to <math>C</math> with respect to <math>AE.</math>
  13 KB (2,298 words) - 12:56, 10 September 2023
• Mock AIME I 2012 Problems/Problem 7
  ...math>P_{O_1}(P), P_{O_2}(P)</math> denote the power of <math>P</math> with respect to the circles <math>O_1, O_2</math>, respectively. Then <math>P_{O_1}(P) =
  2 KB (380 words) - 17:38, 7 April 2012
• 2012 USAJMO Problems/Problem 1
  ...nce <math>AP = AQ</math>, <math>AP^2 = AQ^2</math>, so the power of A with respect to both circumcircles is the same. Thus, <math>A</math> lies on the [[Radic
  4 KB (613 words) - 20:50, 19 December 2023

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