Length of Median?

Geometry Level 5

$\triangle ABC$ has an area of $15\sqrt{3}$ with $\angle BAC =120^{\circ}$ and $\angle ACB < \angle ABC$ . The distance from $A$ to the centre of the circle inscribed in $\triangle ABC$ is $2$ .

If the length of the median drawn from $B$ is $\sqrt{ab}$ , where $a$ and $b$ are positive coprime integers and square-free, find $a + b$ .

The answer is 20.

2 solutions

Ahmad Saad
Jun 5, 2016

Nicely done. +1!

Rishabh Tiwari - 5 years ago

Rishabh Tiwari
Jun 6, 2016

@Deeparaj Bhat , can this problem be solved using complex numbers? I really like your method bhaiya!, @Ashish Siva plz comment , this problem is one of my favourites!

You could, if you wanted. Taking the circumcentre as origin, the expression for the incentre is given in the third book (about complex numbers) that I've given you. But it'll be very tedious and quite artificial. This problem is more suited to trigonometric methods, as illustrated by the solution above.

A Former Brilliant Member - 5 years ago

Ok if thats what you say, thanks for your help, I look forward to see a good solution, an interesting one , to this problem!

Rishabh Tiwari - 5 years ago

Length of Median?

The answer is 20.

2 solutions

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