TITLE: Problem finding coordinates in a earth like coordination system
QUESTION [0 upvotes]: A picture with the problem
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.

REPLY [0 votes]: 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.
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