Girls in Math at Yale 2021 Problem 7: Circles!!!!!!!
Source:
February 27, 2021
Yalecollege
Problem Statement
Suppose two circles and with centers and have radii and , respectively. Suppose that points and lie on circles and , respectively, such that segments and intersect and that is tangent to and . If , find the area of quadrilateral .
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(12cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179; /* image dimensions */ /* draw figures */
draw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2));
draw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2));
draw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2));
/* dots and labels */
dot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle);
label("", (-4.252707018231291,-1.545940604327141), N * labelscalefactor);
dot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle);
label("", (21.704189347819636,-1.250863978037686), NE * labelscalefactor);
dot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle);
label("", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor);
dot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle);
label("", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]Proposed by Deyuan Li and Andrew Milas