VMO 1984 Problem 6

(1) ABCD is a tetrahedron. ∠CAD = z, ∠BAC = y, ∠BAD = x, the angle between the planes ACB and ACD is X, the angle between the planes ABC and ABD is Z, the angle between the planes ADB and ADC is Y. Show that sin x/sin X = sin y/sin Y = sin z/sin Z and that x + y = $180^o$ iff X + Y = $180^o$. (2) ABCD is a tetrahedron with ∠BAC = ∠CAD =∠DAB = $90^o$. Points A and B are fixed. C and D are variable. Show that ∠CBD + ∠ABD + ∠ABC is constant. Find the locus of the center of the insphere of ABCD.