MathDB

Let

\triangle ABC

be an acute triangle with circumcenter

O

and centroid

G

. Let

X

be the intersection of the line tangent to the circumcircle of

\triangle ABC

A

and the line perpendicular to

GO

G

. Let

Y

be the intersection of lines

XG

and

BC

. Given that the measures of

\angle ABC, \angle BCA,

and

\angle XOY

are in the ratio

13 : 2 : 17,

the degree measure of

\angle BAC

can be written as

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m+n

. [asy] unitsize(5mm); pair A,B,C,X,G,O,Y; A = (2,8); B = (0,0); C = (15,0); dot(A,5+black); dot(B,5+black); dot(C,5+black); draw(A--B--C--A,linewidth(1.3)); draw(circumcircle(A,B,C)); O = circumcenter(A,B,C); G = (A+B+C)/3; dot(O,5+black); dot(G,5+black); pair D = bisectorpoint(O,2*A-O); pair E = bisectorpoint(O,2*G-O); draw(A+(A-D)*6--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10)); draw(intersectionpoint(G--G+(G-E)*10,B--C)--intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10)); X = intersectionpoint(G--G+(E-G)*15,A+(A-D)--A+(D-A)*10); Y = intersectionpoint(G--G+(G-E)*10,B--C); dot(Y,5+black); dot(X,5+black); label("

A

",A,NW); label("

B

",B,SW); label("

C

",C,SE); label("

O

",O,ESE); label("

G

",G,W); label("

X

",X,dir(0)); label("

Y

",Y,NW); draw(O--G--O--X--O--Y); markscalefactor = 0.07; draw(rightanglemark(X,G,O)); [/asy]

Problem Statement