Question

In triangle ABC, the angle bisector of angle BAC meets AC at D. If angle BAC = 60, angle ABC = 60, and AD = 24, then find the area of triangle ABC.

Answer 1

The bisector of angle BAC meets BC, not AC. So, if you meant the bisector of angle ABC meets AC at D, then
Since BAC=ABC = 60°, then ACB is also 60°, making ABC an equilateral triangle.
In triangle ABD, angle ABD=30°, so AB=48
The area of an equilateral triangle of side 48 is 48^2/4 √3 = 576√3

Answer 2

Area of triangle ABC = (1/2) * AD * BD
BD = (24 * √3)/2
Area of triangle ABC = (1/2) * 24 * (24 * √3)/2
Area of triangle ABC = 288√3/2