Properties Under Inversion
- Inversive properties of Steiner chains
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In an annular Steiner chain, the angle subtended by a single circle is 2θ (gold lines), which is also the angle subtended by adjacent points of tangency.
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Two circles (pink and cyan) that are internally tangent to both given circles and whose centers are collinear with the center of the given circles intersect at the angle 2θ.
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Under inversion, these lines and circles become circles with the same intersection angle, 2θ. The gold circles intersect the two given circles at right angles, i.e., orthogonally.
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The circles passing through the mutual tangent points of the Steiner-chain circles are orthogonal to the two given circles and intersect one another at multiples of the angle 2θ.
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The circles passing through the tangent points of the Steiner-chain circles with the two given circles are orthogonal to the latter and intersect at multiples of the angle 2θ.
Circle inversion transforms one Steiner chain into another with the same number of circles.
In the transformed chain, the tangent points between adjacent circles of the Steiner chain all lie on a circle, namely the concentric circle midway between the two fixed concentric circles. Since tangencies and circles are preserved under inversion, this property of all tangencies lying on a circle is also true in the original chain. This property is also shared with the Pappus chain of circles, which can be construed as a special limiting case of the Steiner chain.
In the transformed chain, the tangent lines from O to the Steiner chain circles are separated by equal angles. In the original chain, this corresponds to equal angles between the tangent circles that pass through the center of inversion used to transform the original circles into a concentric pair.
In the transformed chain, the n lines connecting the pairs of tangent points of the Steiner circles with the concentric circles all pass through O, the common center. Similarly, the n lines tangent to each pair of adjacent circles in the Steiner chain also pass through O. Since lines through the center of inversion are invariant under inversion, and since tangency and concurrence are preserved under inversion, the 2n lines connecting the corresponding points in the original chain also pass through a single point, O.
Read more about this topic: Steiner Chain
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