▁. ▁ ▁D ao ' s ▁general ization ▁ ▁The ▁theorem ▁was ▁further ▁generalized ▁by ▁D ao ▁Th anh ▁O ai . ▁The ▁general ization ▁as ▁follows : ▁ ▁First ▁general ization : ▁Let ▁ABC ▁be ▁a ▁triangle , ▁P ▁be ▁a ▁point ▁on ▁the ▁plane , ▁let ▁three ▁parallel ▁segments ▁AA ', ▁BB ', ▁CC ' ▁such ▁that ▁its ▁mid points ▁and ▁P ▁are ▁col linear . ▁Then ▁PA ', ▁P B ', ▁PC ' ▁meet ▁BC , ▁CA , ▁AB ▁respectively ▁at ▁three ▁col linear ▁points . ▁▁▁▁ ▁Second ▁general ization : ▁Let ▁a ▁con ic ▁S ▁and ▁a ▁point ▁P ▁ ▁on ▁the ▁plane . ▁Con struct ▁three ▁lines ▁ ▁da , ▁db , ▁d c ▁through ▁P ▁such ▁that ▁they ▁meet ▁the ▁con ic ▁at ▁A , ▁A '; ▁▁ ▁B , ▁B ' ▁ ▁; ▁ ▁C , ▁C ' ▁respectively . ▁Let ▁D ▁ ▁be ▁a ▁point ▁on ▁the ▁polar ▁of ▁point ▁P ▁with ▁respect ▁to ▁( S ) ▁or ▁D ▁lies ▁on ▁the ▁con ic ▁( S ). ▁Let ▁DA ' ▁ ∩ ▁BC ▁= A 0 ; ▁DB ' ▁ ∩ ▁AC ▁= ▁B 0 ; ▁DC ' ▁ ∩ ▁AB = ▁C 0 . ▁Then ▁A 0 , ▁B 0 , ▁C 0 ▁ ▁are ▁col linear . ▁ ▁References ▁▁▁▁ ▁Category : E uclidean ▁geometry ▁Category : Con ic ▁sections ▁Category : The or ems ▁in ▁plane ▁geometry <0x0A> </s> ▁Third ▁All - U k rain ian ▁Congress ▁of ▁S ovi ets ▁()