There is a vector lying in the tangent plane to a sphere $S^2$ at equator. We take two its "stereographic" projections - one from the south pole and other - from north. Projections to the planes tangent to sphere at respectively north and south poles. How to show that the projections of this vector will be symmetric relatively to the projection of equator (i.e. relatively to the tangent line to equator's projection)?

UPD I draw a picture enter image description here

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    $\begingroup$ I think you can take advantage of the fact that stereographic projection is a conformal map: en.wikipedia.org/wiki/Stereographic_projection $\endgroup$ – Holonomia Jul 26 '15 at 16:32
  • $\begingroup$ @Holonomia And the fact that it lies by different sides of the equator projection comes from the fact that large circle projection (equivalent to the initial tangent vector) will be projected to inner and outer part of the equator image? $\endgroup$ – Stan Jul 26 '15 at 16:49
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    $\begingroup$ It seems to me that a tangent vector to the equator is left invariant by the conformal map between the two tangent spaces. So such a conformal map is an isometry. Since it is not the identity the only possibility is a symmetry w.r.t. the fix vector i.e. the tangent to the equator, isn't it?. $\endgroup$ – Holonomia Jul 26 '15 at 17:45

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