Where Do They Concur?

Geometry Level 4

If an acute \(\triangle ABC,\) \(D,E,F\) are the feet of perpendiculars from \(A,B,C\) on \(BC,CA,AB\) respectively. Lines \(EF, ED, DE\) meet lines \(BC,CA,AB\) at points \(P,Q,R\) respectively. Let \(H\) be the orthocenter of \(\triangle ABC.\) Construct lines \(\ell _A, \ell _B, \ell _C\) passing through \(A,B,C\) and perpendicular to \(PH,QH,RH\) respectively. It turns out that lines \(\ell _A, \ell _B, \ell _C\) are concurrent at a point \(X\) within \(\triangle ABC.\) Then, \(X\) is the .......... of \(\triangle ABC.\)

Details and assumptions

  • The picture shows how \(\ell_A\) is defined. Lines \(\ell_B\) and \(\ell_C\) are defined analogously.

  • This problem is inspired from a problem which appeared in the India TST.

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