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length of segment is constant, independent of choice of line (2 circles)

Source: 2016 Peru Cono Sur TST P6

February 26, 2020
circlesfixedSegmentgeometry

Problem Statement

Two circles ω1\omega_1 and ω2\omega_2, which have centers O1O_1 and O2O_2, respectively, intersect at AA and BB. A line ℓ\ell that passes through BB cuts to ω1\omega_1 again at CC and cuts to ω2\omega_2 again in DD, so that points C,B,DC, B, D appear in that order. The tangents of ω1\omega_1 and ω2\omega_2 in CC and DD, respectively, intersect in EE. Line AEAE intersects again to the circumscribed circumference of the triangle AO1O2AO_1O_2 in FF. Try that the length of the EFEF segment is constant, that is, it does not depend on the choice of ℓ\ell.
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