In the figure below, $AB=75,$ $AC=70,$ and $BY=60$. Compute $CX$. [asy]
size(5cm);
pair a=(495/13,840/13); pair b=(0,0); pair c=(65,0);
dot(a); dot(b); dot(c);
pair x=(33/75)*a;
pair y=(5/14)*(a-c)+c;
dot(x); dot(y);
draw(c--b--a--c);
draw(x--c, dashed);
draw(b--y, dashed);
draw(x+(c-x)/11.2--x+(c-x)/11.2+(a- x)/8.4--x+(a-x)/8.4);
draw(y+(c-y)/5--y+(c-y)/5+(b-y)/12- -y+(b-y)/12);
label("$A$",a, N);
label("$B$",b, SW);
label("$C$",c, SE);
label("$X$",x, WNW);
label("$Y$",y, ENE);
[/asy]