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SIX CONCURRENT LINES ON THE NPC

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  • Antreas Hatzipolakis
    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
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    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





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