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Showing trigonometric relation and angle relation.

Source: Vietnam MO 1984 P6

March 20, 2011
trigonometrygeometryincentergeometry unsolved

Problem Statement

Consider a trihedral angle SxyzSxyz with \angle xSz = \alpha , \angle xSy = \beta and ∠ySz=γ\angle ySz =\gamma. Let A,B,CA,B,C denote the dihedral angles at edges y,z,xy, z, x respectively. (a)(a) Prove that sin⁡αsin⁡A=sin⁡βsin⁡B=sin⁡γsin⁡C\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C} (b)(b) Show that α+β=180∘\alpha + \beta = 180^{\circ} if and only if ∠A+∠B=180∘.\angle A + \angle B = 180^{\circ}. (c)(c) Assume that \alpha=\beta =\gamma = 90^{\circ}. Let O∈SzO \in Sz be a fixed point such that SO=aSO = a and let M,NM,N be variable points on x,yx, y respectively. Prove that ∠SOM+∠SON+∠MON\angle SOM +\angle SON +\angle MON is constant and find the locus of the incenter of OSMNOSMN.
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