+359 In triangle ABC, the angle bisector of ∠BAC meets ¯BC at D. If ∠BAC=60∘, ∠ABC=60∘, and AD=24, then find the area of triangle ABC.
+410 
Because BAC = 60, ABC = 60, ACB should equal 60.
Therefore, ABC is equilateral.
Because AD is an angle bisector, then ABD, is 30, and so ADB = ADC = 90.
Therefore, set BD = x.
x2+242=(2x)2.
We get x=8√3. (These triangle ratios are very special, see if you can find the ratios for a 30-60-90 triangle for yourself)
Therefore the area is 8√3∗24∗22=192√3.
+410
Because BAC = 60, ABC = 60, ACB should equal 60.
Therefore, ABC is equilateral.
Because AD is an angle bisector, then ABD, is 30, and so ADB = ADC = 90.
Therefore, set BD = x.
x2+242=(2x)2.
We get x=8√3. (These triangle ratios are very special, see if you can find the ratios for a 30-60-90 triangle for yourself)
Therefore the area is 8√3∗24∗22=192√3.
