In an equilateral triangle ABC, BD is a median. Let us prove that, ∠ ABD = cot ∠BAD

To Prove : cot ∠ ABD = cot ∠BAD

Proof:

In the figure shown above ABC is an equilateral triangle

Now BD is a median on side AC, Therefore, AD = DC

And ∠ ADB = 90°

(By property of equilateral triangle if a median is dropped from one vertex to opposite side, it is perpendicular to the side, and ∠ ABD = ∠BAD = 45°)

If the angles are equal, their values of cot will also be equal

So,

Cot ∠ ABD = cot ∠BAD

Hence, Proved