On the sides BC, CA, AB of the triangle ABC are described
two sets of equilateral triangles,
the set Ba′C, Cb′A, Ac′B externally, and
the set BaC, CbA, AcB internally.
The lines Aa′, Bb′, Cc′ cointersect in Q, the centre of Perspective
of the triangles ABC, a′b′c′,
and the lines Aa, Bb, Cc in P, the centre of Perspective of ABC, abc.
Since a, a′, b, b′, c, c′ are on the perpendicular bisectors of BC, CA, AB, their joins cointersect in the circumcentre, O, which is the centre of Perspective of abc, a′b′c′.