Hey guys
Given:
two coordinates $A(a_1,a_2), M(m_1,m_2)$ ,
the distance between $B$ & $C$ is known as $w, d(B,C) = w$
d(B,M) = d(M,C) where d is the great-circle distance.
The two great-cicles 's intersection is at a right angles(90°).
Find the coordinates for $B(b_1,b_2)\quad C(c_1,c_2).$
The Earth radius is known as R.
Since we are given only one "radius" of the Earth, and the question is tagged "spherical-trigonometry", I suppose we are meant to assume that all these coordinates and measurements are made on a perfectly spherical planet.
So just think the problem through logically, step by step, starting with what you know about the problem and what you can derive from that.
You know the coordinates of $M$ and $A$, so you can derive the distance between them (using the Earth radius $R$), the direction from $A$ to $M$, and the direction from $M$ to $A$, all using well-known formulas.
You know that the two great circles meet at $M$ at a $90$-degree angle, so if you can find the direction from $M$ to any of the three other points then you know the directions from $M$ to each of the remaining two points.
You know that the point $M$ divides the arc from $B$ to $C$ in two equal parts, and you know the length of the whole arc. From this it should be clear how you can derive the distances $d(M,B)$ and $d(M,C)$.
Given the Earth radius $R$, the coordinates of a starting point, and the direction and distance to another point, there are formulas that tell you the coordinates of the second point.
By applying all of these steps one by one to the information you have at an given time, you can derive more information about the problem. Continue doing this until you have the answer.
If you are clever, you will notice that some of the information I said you can derive will not actually be used in deriving any further information, and you can save the trouble of actually computing the unused information.