Stereographic projection

A stereographic projection is a projection from a sphere to a tangent plane. Stereographic projections preserve angles.

To stereographically project a point on a sphere to a plane tangent to its south pole, draw the line from the north pole of the sphere to the point in question. The stereographic projection of this point is then the intersection of this line with the plane. As such, the stereographic projection of the north pole will be undefined.

Moscow Math Olympiad, 1950

A spatial quadrilateral is circumscribed about the sphere.

Prove that the four points of contact lie in one plane.

Proof

Let given quadrilateral be $EFGH,$ the points of contact be $A,B,C,D,$ the north pole of the sphere be point $D.$ Denote sphere as $\Omega.$

Let plane $EFG$ cross sphere at circle $\omega.$ Points $D$ and $C$ lies on $\omega.$ Similarly define circles $\omega_a$ with points $A$ and $D, \omega_b$ with points $A$ and $B, \omega_c$ with points $B$ and $C.$

This circles lies in different planes and have the common points, so they are tangent in pares.

The stereographic projection from point $D$ is shown on diagram. Points $A,B,C$ maps into points $A', B', C',$ tangent circles $\omega_b$ and $\omega_c$ maps into tangent circles $\omega'_b$ and $\omega'_c,$ circles $\omega_a$ and $\omega$ maps into parallel lines $\omega'_a$ and $\omega',$ tangent to circles $\omega'_b$ and $\omega'_c,$ respectively.