Three chords of a sphere are meet at a point X inside the sphere but are not coplanar. A sphere through an endpoint of each chord and X touches the sphere through the other endpoints and X. Show that the chords have equal length.
Solution
Let two of the chords be AB and CD. Take the plane containing them. In this plane we have a circle through A, B, C, D, and a circle through A, C, X which touches a circle through B, D, X at X. We show that AX = CX. Let the common tangent at X meet the larger circle at E and F. [Assume the points are in the order A, C, F, B, D, E as we go round the circle.] We have ∠XAC = ∠BAC (same angle) = ∠BDC (circle ABCD) = ∠BXF (XF tangent to BXD) = ∠EXA (opposite angle) = ∠XCA (EX tangent to AXC). So XAC is isosceles. So XA = XC. Similarly XB = XD. Hence AB = CD.
Similarly, the other pairs of chords are equal.
© John Scholes
jscholes@kalva.demon.co.uk
21 Aug 2002