Spherical Triangles I'm trying to figure out how to calculate coordinates on a globe, and I would like to ask for some help.

Let's say I have POINT A on the globe with the following coordinates:
POINT A
Latitude 45° 27' 50.95" N
Longitude 9° 11' 23.98" E

Also I have POINT B which is the Antipode:
POINT B
Latitude 45° 27' 50.95" S
Longitude 170° 48' 36.02" W

Given we are on a sphere (globe), from Point A to Point B for example I can draw 360 great circles, one for each single degree of the sphere:


*

*and the distance to go from Point A to Point B is 180° (first semi-circle)

*and the distance to go back from Point B to Point A is also 180° (second semi-circle)



Now let's say I have POINT C with the following coordinates:
POINT C
Latitude 45° 26' 48.53" N
Longitude 9° 1' 58.11" E

Given these information, THERE IS ONLY ONE GREAT CIRCLE which:
START IN POINT A
GOES THROUGH POINT C
ARRIVE IN POINT B (completing the first semi-circle of 180°)
COME BACK IN POINT A (completing the second semi-circle of 180°)

My problem is to find the formula to calculate the coordinates of the 2 Points which are half-way (90°) from Point A to Point B.
Let's call these 2 Points as M and N:
START in POINT A
GOES THROUGH POINT C
PASS THROUGH POINT M (at 90°)
ARRIVE IN POINT B (completing the first semi-circle of 180°)
PASS TO POINT N (at 270°)
COME BACK IN POINT A (completing the second semi-circle of 180°)
--- Which is the formula to calculate M and N?

In a - plain surface - i would have used the simple proportion of triangles to calculate them, but given is a sphere I don't know the formula to be applied. 
I did some online search but I find a kind of difficult to figure it out. 
I have been out of school from 15 years now, so I would like to ask if somebody can help me.
I hope my explanation is clear, thx a lot if you can help!!!
Cheers
 A: This answer is not pretty, but it will get the job done.  Also, there will be some error since the Earth is an ellipsoid and not a perfect sphere.
First, convert the points $A$ and $C$ from (longitude, latitude) to points on a sphere.  Let $\theta$ be the angle that describes your longitude $\mathit{east}$ of the prime meridian.  If your longitude is measured as west of the prime meridian, then make it negative.  (For example, change $34^\circ$W to $-34^\circ$.  Let $\phi$ be equal to $90^\circ$ minus your latitude $\mathit{north}$ of the equator or $90^\circ$ plus your latitude $\mathit{south}$ of the equator.  You now have your points $A$ and $C$ described by two angles without the pesky NESW signifiers.  $A = (\theta_A, \phi_A)$ and $C = (\theta_C, \phi_C)$.
Second, convert $A$ and $C$ to points in 3-dimensional space.  To do this:
$$A = \left< A_x, A_y, A_z \right> = \left< \cos \theta_A \sin \phi_A,\; \sin \theta_A \sin \phi_A,\; \cos \phi_A \right>$$
$$C = \left< C_x, C_y, C_z \right> = \left< \cos \theta_C \sin \phi_C,\; \sin \theta_C \sin \phi_C,\; \cos \phi_C \right>$$
Third, we find the 3-dimensional coordinates $M$ using the Gram-Schmidt process.  To understand this process, you might want to check out vector dot product and vector norm.  We will do this step in two parts.  First, we will find $M'$, then we will scale it to find $M$.  The vector formula for $M'$ is
$$M' = C - (C \cdot A)A$$
Thus, the components of $M' = \left< M'_X, M'_Y, M'_Z \right>$ are
$$M'_X = C_X - (C_XA_X + C_YA_Y + C_ZA_Z)A_X$$
$$M'_Y = C_Y - (C_XA_X + C_YA_Y + C_ZA_Z)A_Y$$
$$M'_Z = C_Z - (C_XA_X + C_YA_Y + C_ZA_Z)A_Z$$
Now we find $M$ by scaling each of these components by dividing by $||M'||$.  For $M = \left< M_X, M_Y, M_Z \right>$ we have
$$M_X = \frac{M'_X}{\sqrt{(M'_X)^2 + (M'_Y)^2 + (M'_Z)^2}}$$
$$M_Y = \frac{M'_Y}{\sqrt{(M'_X)^2 + (M'_Y)^2 + (M'_Z)^2}}$$
$$M_Z = \frac{M'_Z}{\sqrt{(M'_X)^2 + (M'_Y)^2 + (M'_Z)^2}}$$
Finally, we convert $M$ into longitude and latitude.
$$\phi_M = \arccos(M_Z)$$
$$\tan \theta_M = \frac{M_Y}{M_X}$$
Be careful with $\theta_M$, it is either $\arctan \left(\frac{M_Y}{M_X}\right)$ or $\arctan \left(\frac{M_Y}{M_X}\right) + 180^\circ$.  Then convert $\theta_M$ and $\phi_M$ into longitude and latitude respectively.  The point $N$ is just the antipode of $M$.
