Δευτέρα 21 Οκτωβρίου 2019

HYACINTHOS 23128

Antreas P. Hatzipolakis
 

Let ABC be a triangle with orthocenter H, incenter I and excenters Ia,Ib,Ic.
A',B',C' lie on BC,CA,AB, resp. such that  B'C' _|_ IaH, C'A' _|_ IbH, A'B' _|_ IbH.
Prove that the NPC center of the triangle A'B'C' lies on the line IH.

References:
Buratino Giggle [=Tran Quang Hung]
https://www.facebook.com/groups/439719556180075/permalink/444311072387590/
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=48&t=625851

Variation
 
The perpendiculars to HIa, HIb, HIc from A,B,C bound triangle A"B"C".
Which point is the circumenter of A"B"C" ?
Is it lying on the IH line?
 
APH

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