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  • ...First let the area of the circle be <math> A </math> and the area of the triangle be <math> T </math>. We have three cases then. '''Case 1:''' The circle's area is greater than the triangle's area.
    9 KB (1,585 words) - 12:46, 2 September 2024
  • ...c}</math> determine two angles of a triangle. What is the probability this triangle is obtuse? ...ega_{a}</math> and <math>\omega_{b}</math> with radius <math>1</math> have centers that are <math>\tfrac{4}{3}</math> units apart. Two externally tangent circ
    12 KB (1,784 words) - 15:49, 1 April 2021
  • ...th of the third side is 15. What is the greatest possible perimeter of the triangle? Circles with centers <math> O</math> and <math> P</math> have radii 2 and 4, respectively, and a
    13 KB (2,058 words) - 11:36, 4 July 2023
  • The vertices of a <math>3-4-5</math> right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of Circles with centers <math>A</math> and <math>B</math> have radii <math>3</math> and <math>8</ma
    15 KB (2,223 words) - 12:43, 28 December 2020
  • ...gle <math>BAC</math>. Which of the following is closest to the area of the triangle <math>ADE</math>? ...the area of the square. The [[ratio]] of the area of the other small right triangle to the area of the square is
    13 KB (1,948 words) - 09:35, 16 June 2024
  • ...> and <math>BG</math> intersect at <math>E</math>. Find the area of <math>\triangle AEB</math>. radius 2. The centers of the small semicircles divide AB into four line segments
    13 KB (1,987 words) - 17:53, 10 December 2022
  • Circles with centers <math> O</math> and <math> P</math> have radii 2 and 4, respectively, and a ...a rectangle with width <math>2</math> and base <math>x</math>, and a right triangle with one leg <math>2</math>, the hypotenuse <math>6</math>, and the other <
    3 KB (458 words) - 15:40, 6 October 2019
  • The [[vertex|vertices]] of a <math>3-4-5</math> [[right triangle]] are the centers of three mutually externally tangent [[circle]]s, as shown. What is the sum
    1 KB (184 words) - 12:57, 19 January 2021
  • Circles with centers <math>A</math> and <math>B</math> have radius 3 and 8, respectively. A [[co ...int on a circle are always perpendicular). Thus, <math>\triangle ACE \sim \triangle BDE</math>.
    2 KB (286 words) - 09:16, 19 December 2021
  • ...ach chosen from the set <math>\{0,1,2\}</math>. How many [[equilateral]] [[triangle]]s all have their [[vertices]] in <math>S</math>? ...uilateral triangles. We have 8 unit cubes, and then the entire cube (green triangle), giving us 9 cubes and <math>9 \cdot 8 = 72</math> equilateral triangles.
    4 KB (495 words) - 00:36, 26 May 2024
  • ...t{3} + 1</math>. To verify this is the longest length, we can see from the triangle inequality that the length from the origin to any other point on the sphere ...hese <math>4</math> circles, notice that if you connect the <math>4</math> centers as a square, the diameter of the large circle will be the diagonal of the s
    2 KB (364 words) - 03:54, 16 January 2023
  • ...and <math>\overline{AC}</math> are congruent. What is the area of <math>\triangle ABC</math>? Circles with centers <math>A</math> and <math>B</math> have radii <math>3</math> and <math>8</ma
    13 KB (2,028 words) - 15:32, 22 March 2022
  • ...and <math>\overline{AC}</math> are congruent. What is the area of <math>\triangle ABC</math>? Let the centers of the smaller and larger circles be <math>O_1</math> and <math>O_2</math>
    5 KB (811 words) - 10:44, 30 November 2024
  • ...h> C_1 </math> and <math> C_2 </math> are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of <math> C_3 </math> is al ...gular octahedron, that <math> C </math> is the cube whose vertices are the centers of the faces of <math> O, </math> and that the ratio of the volume of <math
    7 KB (1,119 words) - 20:12, 28 February 2020
  • ...r [[right triangle]]s <math>\triangle HO_1T_1 \sim \triangle HO_2T_2 \sim \triangle HO_3T </math>, ..._3T = \frac{58}{7}\dagger</math>. By the [[Pythagorean Theorem]] on <math>\triangle ATO_3</math>, we find that
    4 KB (693 words) - 12:03, 28 December 2021
  • ...<math> C </math> is the [[cube (geometry) | cube]] whose vertices are the centers of the faces of <math> O, </math> and that the ratio of the volume of <math ...io is <math>\frac 23</math> (because the [[triangle median |medians]] of a triangle are trisected by the centroid), so <math>GH = \frac{2}{3}MN = \frac{2s}3</m
    3 KB (436 words) - 19:35, 13 August 2024
  • ...d <math>r_2</math> are externally tangent, then the distance between their centers is <math>r_1 + r_2</math>, and if they are internally tangent, it is <math> ...2=20</math>. In particular, the locus of points <math>C</math> that can be centers of circles must be an ellipse with foci <math>F_1</math> and <math>F_2</mat
    12 KB (2,001 words) - 19:26, 23 July 2024
  • ...ith [[hypotenuse]] one unit away from <math>\overline{AC}</math>. Let this triangle be <math>A'B'C'</math>. ...of <math>\triangle ABC</math> extend one unit farther than those of <math>\triangle A'B'C'</math>. From <math>A = rs</math>, we note that <math>r_{ABC} = \frac
    5 KB (836 words) - 06:53, 15 October 2023
  • ...<math>9</math>, and <math>49</math>, respectively. Find the area of <math>\triangle ABC</math>. Three circles, each of radius <math>3</math>, are drawn with centers at <math>(14, 92)</math>, <math>(17, 76)</math>, and <math>(19, 84)</math>.
    6 KB (933 words) - 00:15, 19 June 2022
  • ...th> The [[ratio]] of the area of triangle <math>ABC</math> to the area of triangle <math>PQR</math> is <math>a + b\sqrt {c},</math> where <math>a, b</math> an
    7 KB (1,084 words) - 01:01, 28 November 2023

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