Mock AIME 1 2007-2008 Problems/Problem 11

Problem

is inscribed inside such that lie on , respectively. The circumcircles of have centers , respectively. Also, , and . The length of can be written in the form , where and are relatively prime integers. Find .

Solution

size(150); defaultpen(linewidth(0.8)); import markers; import geometry_dev;
pair B = (0,0), C = (25,0), A = (578/50,19.8838);
draw(A--B--C--cycle);
label("$B$",B,SW); label("$C$",C,SE); label("$A$",A,N);
pair D = (13,0), E = (11*A + 13*C)/24, F = (12*B + 11*A)/23;
draw(D--E--F--cycle);
label("$D$",D,dir(-90));
label("$E$",E,dir(0));
label("$F$",F,dir(180));

draw(A--E,StickIntervalMarker(1,3,size=6));draw(B--D,StickIntervalMarker(1,3,size=6));
draw(F--B,StickIntervalMarker(1,2,size=6)); draw(E--C,StickIntervalMarker(1,2,size=6));
draw(A--F,StickIntervalMarker(1,1,size=6)); draw(C--D,StickIntervalMarker(1,1,size=6));

label("24",A--C,5*dir(0)); label("25",B--C,5*dir(-90)); label("23",B--A,5*dir(180));
 (Error making remote request. Unknown error_msg)

From adjacent sides, the following relationships can be derived:

$\begin{align*} DC &= EC + 1\\ AE &= AF + 1\\ BD &= BF + 2 \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Since , and , . Thus, . . Thus, . Thus, the answer is .

See also

Mock AIME 1 2007-2008 (Problems, Source)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15