Where Do They Concur?

Geometry Level 3

Let \(\triangle ABC\) be an acute triangle with circumcircle \(\omega.\) \(D,E,F\) are the feet of perpendiculars from \(A,B,C\) to \(BC,CA,AB\) respectively. Let \(\ell_A, \ell_b, \ell_C\) be the lines passing through \(A,B,C\) and parallel to \(BC,CA,AB\) respectively. Let \(P,Q,R\) be the second points of intersection (apart from \(A,B,C\) respectively) of \(\ell_A, \ell_B, \ell_C\) with \(\omega\) respectively. It turns out that lines \(PD, QE, RF\) concur at a point \(X\) within \(\triangle ABC.\) Then, \(X\) is the ............ of \(\triangle ABC.\)

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