2018 AIME I Problems/Problem 4
Problem 4
In 
and 
. Point 
lies strictly between 
and 
on 
and point 
lies strictly between and 
on 
) so that 
. Then 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
Solution 1
![[asy] import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair B = (0,0), A = (6,8), C = (12,0), D = (2.154,2.872), E = (8.153, 5.128), F=(7.68,5.76), G=(7.077,6.564), H=(5.6,4.3), I=(4.5,6), J=(10,2.66); pair[] dotted = {A,B,C,D,E,F,G}; D(A--B); D(C--B); D(A--C); D(D--E); pathpen=dashed; D(B--F); D(D--G); dot(dotted); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,NE); label("$F$",F,NE); label("$G$",G,NE); label("$x$",H,NW); label("$x$",I,NW); label("$x$",J,NE); [/asy]](http://latex.artofproblemsolving.com/e/1/f/e1f19761efd72d9c597461c3953314009f91a01a.png)
