locus of projection of a point on plane, locus of circumcenter of sphere

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locus of projection of a point on plane, locus of circumcenter of sphere

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Source: Spanish Mathematical Olympiad 1994 P2

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#1 h Jul 31, 2018, 10:57 AM • 2 Y

Y by Adventure10, Mango247

Let $Oxyz$ be a trihedron whose edges $x,y, z$ are mutually perpendicular. Let $C$ be the point on the ray $z$ with $OC = c$ . Points $P$ and $Q$ vary on the rays $x$ and $y$ respectively in such a way that $OP+OQ = k$ is constant. For every $P$ and $Q$ , the circumcenter of the sphere through $O,C,P,Q$ is denoted by $W$ . Find the locus of the projection of $W$ on the plane O $xy$ . Also find the locus of points $W$ .

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#2 h Aug 8, 2018, 2:13 PM • 2 Y

Y by Adventure10, Mango247

The locus of the projection of $W$ on the plane O $xy$ is the line $x+y=\frac{k}{2}$ .

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