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Locus of sphere cutting three spheres along great circles
Miquel-point   1
N Apr 7, 2025 by kiyoras_2001
Source: Romanian IMO TST 1981, Day 2 P3
Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$.

Stere Ianuș
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Miquel-point
Apr 6, 2025
kiyoras_2001
Apr 7, 2025
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Locus of sphere cutting three spheres along great circles
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Source: Romanian IMO TST 1981, Day 2 P3
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Miquel-point
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Miquel-point
#1 h  Apr 6, 2025, 6:29 PM • 2 Y
Y by PikaPika999, kiyoras_2001
Consider three fixed spheres $S_1, S_2, S_3$ with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each $S_i$ along a great circle of $S_i$.

Stere Ianuș
This post has been edited 1 time. Last edited by Miquel-point, Apr 6, 2025, 6:30 PM
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kiyoras_2001
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kiyoras_2001
#2 h  Apr 7, 2025, 7:40 PM
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Let $O_i$ and $R_i$ be the center and radius of $S_i$. Choose the point $P_3$ on the segment $O_1O_2$ so that $O_1P_3^2-P_3O_2^2 = R_2^2-R_1^2$. Similarly define $P_1, P_2$.
Consider any sphere $S$ with center $O$ and radius $R$ intersecting all $S_i$ in great circles $\Omega_i$. Then $OO_i$ is perpendicular to the plane of $\Omega_i$ and $OO_i^2 + R_i^2 = R^2$. Hence $O_1O^2 - OO_2^2 = R_2^2-R_1^2 = O_1P_3^2 - O_1P_2^2$ and $OP_3\perp O_1O_2$. Similarly $OP_1\perp O_2O_3$ and $OP_2\perp O_3O_1$. Since $O_1P_3^2 - P_3O_2^2 + O_2P_1^2 - P_1O_3^2 + O_3P_2^2 - P_2O_1^2 = 0$ by Steiner's theorem $O$ exists and lies on a line $\ell$ perpendicular to $O_1O_2O_3$. Obviously for any point $O\in \ell $ we can find a respective sphere $S$ intersecting $S_i$ with great circles. Thus the desired locus is the line $\ell$.
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