Spheres and skew lines

Question

G is a given sphere in the space. For any line e that has no common point with G, define the line f as the conjugate of e with respect to G if f joins the points of tangency on the two planes tangent to G passing through e. Show that two lines of the space passing G are skew if and only if their conjugates with respect to G are skew.

Answer 1

I will prove that if two lines $a$ and $b$ are coplanar, then their conjugate lines $a'$ and $b'$ are coplanar too. This is only a first step in your proof, I'll leave to you the rest.

Let $O$ be the center of sphere $\boldsymbol{G}$ and $r$ its radius. First of all observe that the conjugate $a'$ of a line $a$ lies in the plane through $O$ and perpendicular to $a$. If $A$ is the projection of $O$ on $a$, then $a'$ is perpendicular to $OA$ and the intersection between $OA$ and and $a'$ is the point $A'$ such that $$OA'=r^2/OA.$$

Suppose now that line $a$ lies on a plane $\sigma$ and let $H$ be the projection of $O$ onto $\sigma$. By the theorem of three perpendiculars, the projection of $H$ on $a$ is the same as the projection $A$ of $O$ on $a$. Line $a'$ (the conjugate of $a$) lies on plane $OHA$ and intersects line $OH$ at a point $K$. Triangles $OAH$ and $OA'K$ are similar, so we get: $$ OK={OA'\cdot OA\over OH}={r^2\over OH}. $$

This result does not depend on the position of $a$ on $\sigma$: it follows that the conjugates of lines lying on $\sigma$ all pass through the same point $K$, belonging to the perpendicular line from $O$ to $\sigma$ and whose distance from $O$ is given by the above equation.