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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