<p>Rectangles <img alt="$ABCD$" height="13" src="https://latex.artofproblemsolving.com/f/9/e/f9efaf9474c5658c4089523e2aff4e11488f8603.png" width="57" /> and <img alt="$EFGH$" height="12" src="https://latex.artofproblemsolving.com/9/7/3/9735b37ac43861ef9e105bff1711f117917b09ec.png" width="59" /> are drawn such that <img alt="$D,E,C,F$" height="18" src="https://latex.artofproblemsolving.com/b/e/7/be75d0c6c67aa645ba151d1612702d711498f881.png" width="84" /> are collinear. Also, <img alt="$A,D,H,G$" height="18" src="https://latex.artofproblemsolving.com/f/7/5/f755021bbd34a29b8a0b79947e16401f31f91158.png" width="85" /> all lie on a circle. If <img alt="$BC=16$" height="14" src="https://latex.artofproblemsolving.com/6/f/e/6febf03da2abc694a9cfbe9c1e3ac7caa2abcdf0.png" width="73" />,<img alt="$AB=107$" height="14" src="https://latex.artofproblemsolving.com/7/8/8/7889e5e4034d69c1902fa203b1ef41571be40d09.png" width="82" />,<img alt="$FG=17$" height="14" src="https://latex.artofproblemsolving.com/8/b/a/8ba7d422d4f482f8c194d96cc4303b6a3f241508.png" width="73" />, and <img alt="$EF=184$" height="14" src="https://latex.artofproblemsolving.com/4/c/8/4c8c14036850f199ad2717e318eb4c0c59415a71.png" width="82" />, what is the length of <img alt="$CE$" height="12" src="https://latex.artofproblemsolving.com/1/3/c/13c970cbf5e2e7029089517d5e8efbb8cade6d8a.png" width="28" />?</p> <p><img alt="[asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); dot(A^^B^^C^^D^^E^^F^^G^^H); label("$A$", A, S);label("$B$", B, S);label("$C$", C, N);label("$D$", D, N);label("$E$", E, S);label("$F$", F, S);label("$G$", G, N);label("$H$", H, N); draw(E--D--A--B--C--E--H--G--F--C); [/asy]" height="202" src="https://latex.artofproblemsolving.com/1/2/f/12f0a4224f27ffa406ded2c01cfd349ecd0c04de.png" width="448" /></p>

Question No: 420

Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16$ , $AB=107$ , $FG=17$ , and $EF=184$ , what is the length of $CE$ ?

$[asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); dot(A^^B^^C^^D^^E^^F^^G^^H); label("$A$", A, S);label("$B$", B, S);label("$C$", C, N);label("$D$", D, N);label("$E$", E, S);label("$F$", F, S);label("$G$", G, N);label("$H$", H, N); draw(E--D--A--B--C--E--H--G--F--C); [/asy]$

Editorial

Tags: Geometry

Question Details