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# geometry

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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.

Feb 21, 2023

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Note that $$\triangle ABC$$ is equilateral, so angle bisector $$AD$$ is also an altitude of a 30-60-90 triangle.

This means that the hypotenuse is $$24 \times {2 \sqrt3 \over 3} = 16 \sqrt 3$$

So the area of the triangle is $$(16 \sqrt 3)^2 \times {\sqrt3 \over 4} = \color{brown}\boxed{192 \sqrt 3}$$

Feb 21, 2023