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- ...to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree12 KB (1,665 words) - 06:00, 8 July 2024
- ...to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree1 KB (238 words) - 10:29, 24 July 2024
- <math>10</math> lines and <math>10</math> circles divide the plane into at most <math>n</math> disjoint regions. Compute <mat7 KB (1,094 words) - 15:39, 24 March 2019
- Connect the centers of the tangent circles! (call the center of the large circle <math>C</math>) Notice that we don't even need the circles anymore; thus, draw triangle <math>\Delta ABP</math> with cevian <math>PC</13 KB (1,982 words) - 17:12, 20 December 2022
- ...idpoint of <math>\overline{BC}</math>. What is the sum of the radii of the circles inscribed in <math>\triangle ADB</math> and <math>\triangle ADC</math>?12 KB (1,930 words) - 20:23, 9 September 2022
- ...idpoint of <math>\overline{BC}</math>. What is the sum of the radii of the circles inscribed in <math>\triangle ADB</math> and <math>\triangle ADC</math>? ...using Heron’s formula. We can use that area to find the inradius of the circles by the inradius formula <math>A=sr.</math> Therefore, we get <math>\boxed{\4 KB (601 words) - 00:34, 8 August 2023
- ...cles <math>2</math> and <math>3</math> must be the same, and the colors of circles <math>4</math> and <math>6</math> must be the same. This gives us <math>4</ ...th>4</math>, and <math>6</math> will be the same. Similarly, the colors of circles <math>2</math>, <math>3</math>, and <math>5</math> will be the same. This i7 KB (1,057 words) - 23:27, 27 August 2022
- .../math>. The line passing through the two points of intersection of the two circles has equation <math>x + y = c</math>. What is <math>c</math>?15 KB (2,343 words) - 18:26, 25 December 2020
- .../math>. The line passing through the two points of intersection of the two circles has equation <math>x+y=c</math>. What is <math>c</math>? The equations of the two circles are <math>(x+10)^2+(y+4)^2=169</math> and <math>(x-3)^2+(y-9)^2=65</math>.2 KB (366 words) - 13:54, 15 February 2021
- Two circles of radius <math>5</math> are externally tangent to each other and are inter14 KB (2,171 words) - 21:10, 4 November 2023
- Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <m1 KB (193 words) - 09:01, 30 July 2024
- ...in the figure below. There is one point <math>B</math> inside all of these circles. When <math>r = \frac{11}{60}</math>, the distance from the center <math>C_7 KB (1,200 words) - 15:02, 8 September 2020
- ...in the figure below. There is one point <math>B</math> inside all of these circles. When <math>r = \frac{11}{60}</math>, the distance from the center <math>C_6 KB (908 words) - 02:35, 23 January 2024
- ...rsion can be a very useful tool in solving problems involving many tangent circles and/or lines. ...h>. Extend the three semicircles to full circles. Label the resulting four circles as shown in the diagram:16 KB (2,516 words) - 23:48, 15 January 2024
- ...e measure, but they point to the same line <math>BC</math>! Hence, the two circles must be congruent. (This is also a well-known result) ...bisector, it must also hit the circle at the point <math>P</math>. The two circles are congruent, which implies <math>MN=MP\implies ND=DP\implies</math> NDP i6 KB (968 words) - 12:05, 7 June 2024
- ...e has radius 24, a second circle has radius 15, and the centers of the two circles are 52 units apart. A line tangent to both circles crosses the line connecting the two centers at a point7 KB (1,192 words) - 15:14, 20 August 2020
- ...e has radius 24, a second circle has radius 15, and the centers of the two circles are 52 units apart. A line tangent to both circles crosses the line connecting the two centers at a point1 KB (211 words) - 18:16, 16 January 2023
- ...math> and <math>N</math>. Let <math>AB</math> be the line tangent to these circles at <math>A</math> and <math>B</math>, respectively, so that <math>M</math> ...ct <math>AB</math> at <math>X</math>. By our lemma, <math>\textit{(the two circles are tangent to AB)}</math>, <math>X</math> bisects <math>AB</math>. Since <3 KB (465 words) - 15:12, 21 June 2024
- ...ne{AB}</math> is divided at <math> C</math> so that <math> AC=3CB</math>. Circles are described on <math> \overline{AC}</math> and <math> \overline{CB}</math ...e radius of the small circle. Draw the line from the center of each of the circles to the point of contact of the tangent of the circle. By similar triangles,1 KB (184 words) - 17:31, 19 September 2022
- ==Solution 2 - Circles== ...th>20,</math> that each contribute <b>two</b> integer lengths (since these circles intersect the hypotenuse twice) from <math>B</math> to <math>\overline{AC}<3 KB (541 words) - 03:32, 23 January 2023