## SIX CONCURRENT LINES ON THE NPC

Expand Messages
• 1. Let ABC be a triangle, La,Lb,Lc three parallel lines passing through A,B,C, resp. and Ma,Mb,Mc the three lines perpendiculars to La,Lb,Lc at A,B,C, resp.
Message 1 of 1 , Dec 28, 2013
• 235 KB
1. Let ABC be a triangle, La,Lb,Lc three parallel lines passing
through A,B,C, resp. and Ma,Mb,Mc the three lines perpendiculars to
La,Lb,Lc at A,B,C, resp.

Denote:

Pab = La /\ Mb, Pba = Lb /\ Ma

Pbc = Lb /\ Mc, Pcb = Lc /\ Mb

Pca = Lc /\ Ma, Pac = La /\ Mc

The lines PabPba, PbcPcb, PcaPac concur
on the NPC.

Reference:
I. Panakis: 2500 Problems of Geometric Loci With Their
Solutions [in Greek]. Athens, ca 1965, p. 654, #582.
2. Let Lh a line passing through through the orthocenter
H  and Mh the line perpendicular to Lh at H.

Denote:

Pah = La /\ Mh, Pha = Lh /\ Ma

Pbh = Lb /\ Mh, Phb = Lh /\ Mb

Pch = Lc /\ Mh, Phc = Lh /\ Mc

The lines PahPha, PbhPhb, PchPhc concur at a point
on the NPC

If the line Lh is parallel to lines La,Lb,Lc then the two points
on the NPC coincide.

APH

Your message has been successfully submitted and would be delivered to recipients shortly.