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LX and BC meet at given circle, perpendularity wanted

Source: 2019 Nigerian Senior Mathematics Olympiad Round 4 (final) problem 2

September 9, 2019
geometrymidpointperpendicularcircle

Problem Statement

Let K,L,MK,L, M be the midpoints of BC,CA,ABBC,CA,AB repectively on a given triangle ABCABC. Let Γ\Gamma be a circle passing through BB and tangent to the circumcircle of KLMKLM, say at XX. Suppose that LXLX and BCBC meet at Γ\Gamma . Show that CXCX is perpendicular to ABAB.
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