Problem:
In triangle ABC, angles A and B measure 60 degrees and 45 degrees, respectively. The bisector of angle A intersects BC at T, and AT=24. The area of the triangle ABC can be written in the form a+bc​, where a,b, and c are positive integers, and c is not divisible by the square of any prime. Find a+b+c.
Solution:
Note that angles C and ATC each measure 75∘, so AC=AT=24. Draw altitude CH of triangle ABC. Then triangle ACH is 30∘−60∘−90∘ and triangle BHC is 45∘−45∘−90∘. Now AH=12 and BH=CH=123​. The area of triangle ABC is thus (1/2)123​(12+123​)=216+723​.
