ABC is an acute triangle with \angle BCA = 35 ^\circ. Denote the circumcenter of ABC as O and the orthocenter of ABC as H. If AO=AH, what is the value of \angle ABC (in degrees)?

To find the value of angle ABC, we can use the given information that AO = AH.

Step 1: Draw triangle ABC with circumcenter O and orthocenter H. Label angle BCA as 35 degrees.

Step 2: Since AO = AH, triangle AOH is an isosceles triangle with AO = AH.

Step 3: In an isosceles triangle, the altitude from the vertex angle bisects the base.

Step 4: Draw the altitude from the vertex angle C to the base AB. Let's call the intersection point of the altitude and AB as D.

Step 5: Since the altitude AD bisects the base AB, we have AD = BD.

Step 6: Now we have two congruent triangles ACD and BCD (by SAS congruence since angle ACD = angle BCD, AD = BD, and CD = CD).

Step 7: From congruent triangles, we can conclude that angle ACD = angle BCD.

Step 8: Sum of the angles in triangle ABC is 180 degrees. So, angle ABC = (180 - angle BCA - angle ACD).

Step 9: Substituting the given angle BCA = 35 degrees and angle ACD = angle BCD, we have angle ABC = (180 - 35 - (angle BCD)).

Since angle ACD = angle BCD, we can rewrite this as angle ABC = (180 - 35 - angle ACD).

Step 10: However, since we don't have the exact value of angle ACD, we cannot find the exact value of angle ABC.

Therefore, without further information about the triangle ABC or any additional angles, we cannot determine the exact value of angle ABC.