Question

Three triangles are drawn at random to a given circle: show that the odds are $3$ to $1$ against the circle being inscribed in the triangle formed by them.

Answer 1

Draw three random lines $P,Q,R,$ in the same place as the circle, and draw to the circle the six tangents parallel to these lines. Then of the $8$ triangles so formed it is evident that the circle will be escribed to $6$ and inscribed in $2$; and as this is true whatever be the original directions of $P,Q,R,$ the required result follows.