QUESTION: Consider a sphere with radius 1 and centre $O$. There are three points on the surface of the sphere, $A$, $B$ and $C$. A vector $\textbf{v}$ is perpendicular to the plane through $A$, $B$ and $C$. The angle between $\overrightarrow{OA}$ and $\textbf{v}$ is $\alpha$, the angle between $\overrightarrow{OB}$ and $\textbf{v}$ is $\beta$ and the angle between $\overrightarrow{OC}$ and $\textbf{v}$ is $\gamma$. Find a relation between $\alpha$, $\beta$ and $\gamma$.
A Note: Although cross products could be incredibly useful, I'm looking for a method without using them. If it cannot be done without them, I'll be happy to accept an answer with it, although it't not ideal.
My attempts:
Considering right triangles with a side as the altitude and equating the expressions for the altitudes, although I have no clue where to go from there.
I also considered finding equations that related the magnitudes of the vectors to the altitude with the edges of the tetrahedron between the points $A$, $B$, $C$, $O$, although I also couldn't get anywhere with that.
Any help or guidance would be appreciated!
Best Answer
Take the plane of $ABC$ to be the $x-y$ plane, and let the centre of the sphere be $P$ on the $z$-axis. $O$ is now the origin. Then $A,B,C$ are three points on the plane.
Consider triangles $OPA$, $OPB$, $OPC$. Their hypotenuse have length 1, so $AO=1=BO=CO$. Then $\cos\alpha=OP/AO=OP/BO=\cos\beta$. Hence the angles are all equal, $\alpha=\beta=\gamma$.