In triangle ABC, we have angle BAC = 60 degrees and angle ABC = 45 degrees. The bisector of angle A intersects line BC at point T, and AT = 24. What is the length of side BC?
Angle ACB = 180 - 60 - 45 = 75
And if AT bisects CAB ....then angles CAT and BAT = 30
Using the Law of Sines
CT = 24 sin (30°) / sin (75°) = 12/sin (75°) = 12csc(75°)
TB = 24 sin (30°) / sin (45°) = 12 / sin (45°) = 12csc (45°)
So CB = CT + TB = 12 ( csc (75°) + csc (45°) ) ≈ 29.39