I've got a bit of a problem. My problem is : on a sphere, show that if you have a quadrilateral from space tangent to it, the points of tangency (or the points of intersection between the sphere and the quadrilateral) are coplanar.
In the beginning, I thought there might be one, two, three or four points of tangency but a teacher told me that in this problem there must be four point of tangency. My teacher also told me that our problem can be decomposed in three problems : the main problem that I told you in the beginning and 2 special cases.
The first special case is if the quadrilateral is plane, the intersection between the quadrilateral and the sphere would be a circle. The second case is more difficult so I think I have to solve the other problems before.
To solve this problem, I have many ideas but I don't know how to do it and if it works. First I think that we can study if the lines who are passing through the points of tangency are secant at one moment. Am I wright ? Can it work ?
Next I used the equation of a sphere which I simplified by taking the radius equal to 1. I also used the fact that 4 points are coplanar if the vector $$AD = n \vec{AB} + m\vec{AC}.$$ Then I put that expression according to $x(A)$, $y(A)$ and $z(A)$ and I put $ x(A)$, $y(A)$ and $z(A)$ in the equation of a sphere in $$A : x(A)^2 + y(A)^2 + z(A)^2 = 1.$$ At the end I find that the equation of the sphere in $A$ depends of the equation of the sphere in $B, C$, and $D$ but I don't know what I can do with that. Moreover a friend told that with that I can find condition for $n$ and $m$, but I don't understand what that means.