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Revision as of 17:55, 8 June 2025
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![[asy] size(216); pair A,B,P,Q,X,Y; A = (0,0); B = (25,0); P = (4/25) * (-7,24); Q = B + (11/25) * (-7,24); X = (4/5) * (3,4); Y = B - (11/5) * (3,4); dot(A); dot(B); draw(circle(A,4)); draw(circle(B,11)); draw(P--Q); draw(X--Y); label("$A$",A,W); label("$B$",B,E); label("$C$",P,NNW); label("$D$",Q,NNW); label("$E$",X,ENE); label("$F$",Y,WSW); [/asy]](http://latex.artofproblemsolving.com/3/2/c/32c5b2cf58dc1576aad1349b412700e262b38ae0.png)
![[asy] size(216); draw((0,10)--(0,0)--(10,0)); draw((5,10)--(0,0)--(10,5)); label("$ct$",(0,10),N); label("$x$",(10,0),E); label("$ct'$",(5,10),N); label("$x'$",(10,5),E); dot((0,3),blue); dot((3,6),blue); draw((0,3)--(3,6),blue); label("$ct_1$",(0,3),W,blue); label("$ct_2'$",(3,6),NW,blue); [/asy]](http://latex.artofproblemsolving.com/5/a/a/5aa32b4b39ea87966619720471695320b8335aa8.png)
![[asy] size(216); pair[] nodes; for (int i = 0; i < 10; ++i) { nodes.push(dir(36 * i)); dot(dir(36 * i)); label(string(i), 1.1 * dir(36 * i)); } draw(circle(nodes[0] - 0.1, 0.1), Arrow); draw(nodes[1] -- nodes[7], Arrow); draw(nodes[2] -- nodes[4], Arrow); draw(nodes[3] -- nodes[1], Arrow); draw(nodes[4] -- nodes[8], Arrow); draw(arc(nodes[5] + 0.1, 0.1, 180, 540), Arrow); draw(nodes[6] -- nodes[2], Arrow); draw(nodes[7] -- nodes[9], Arrow); draw(nodes[8] -- nodes[6], Arrow); draw(nodes[9] -- nodes[3], Arrow); [/asy]](http://latex.artofproblemsolving.com/c/3/6/c36b831cdbef37e828698141e8c3cc82781c55c7.png)
![[asy] unitsize (1 cm); pair A, B, C; A = (3,2); B = intersectionpoint(arc(A,4,200,340),C--(10,0)); C = (0,0); draw(B--A--C--(8,0), Arrow(6)); draw(arc(A,4,200,340), dashed); label("$A$", A, N); label("$B$", B, SE); label("$C$", C, NNW); label("$b$", (A + C)/2, NW); label("$c$", (A + B)/2, NE); label("$\theta$", (1,0.3)); [/asy]](http://latex.artofproblemsolving.com/7/7/2/772e91e3fdc45be95d3db714c3200caae877c64a.png)
