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Apoint T is an equiangular point of a collection of localized vectors if all of them are seen from T at an equal oriented angle. It is proved that almost each planar triple of vectors with fixed origins (not all of which coincide) has an equiangular point. As a consequence, it is proved that if a planar triple of vectors has no equiangular points, then their projections to a certain axis are equal.

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