BC is a triangle with ∠BAC=60∘,AB=5 and AC=25. D is a point on the internal angle bisector of ∠BAC such that BD=DC. What is AD^2?

It is not stated that D lies on BC. This assumption is not necessarily true.

To find AD² in triangle ABC, we can use the Angle Bisector Theorem and the Law of Cosines.

First, let's find the length of BC using the Law of Cosines:
BC² = AB² + AC² - 2 * AB * AC * cos(BAC)
BC² = 5² + 25² - 2 * 5 * 25 * cos(60°)
BC² = 25 + 625 - 250
BC² = 400
BC = 20

Since BD = DC, we can divide BC into two equal parts:
BD = DC = BC / 2
BD = DC = 20 / 2
BD = DC = 10

Now, let's use the Angle Bisector Theorem to find the length of AD:
AD / BD = AC / CD
AD / 10 = 25 / 10
AD = 2.5 * 10
AD = 25

Finally, we can find AD²:
AD² = 25²
AD² = 625

Therefore, AD² is equal to 625.