▁theorem ▁and ▁its ▁proof ▁ ▁Theorem ▁Any ▁o val ▁ ▁in ▁a ▁finite ▁p app ian ▁project ive ▁plane ▁of ▁odd ▁order ▁is ▁a ▁non de generate ▁con ic ▁section . ▁ ▁Proof ▁For ▁the ▁proof ▁we ▁show ▁that ▁the ▁o val ▁has ▁property ▁( P 3 ) ▁of ▁the ▁ 3 - point ▁version ▁of ▁Pascal ' s ▁theorem . ▁ ▁Let ▁be ▁ ▁any ▁triangle ▁on ▁ ▁and ▁ ▁defined ▁as ▁described ▁in ▁( P 3 ). ▁ ▁The ▁p app ian ▁plane ▁will ▁be ▁coordin at ized ▁in hom ogene ously ▁over ▁a ▁finite ▁field ▁ ▁, ▁such ▁that ▁and ▁ ▁is ▁the ▁common ▁point ▁of ▁the ▁tang ents ▁at ▁ ▁and ▁. ▁The ▁o val ▁ ▁can ▁be ▁described ▁using ▁a ▁bi jective ▁function ▁: ▁ ▁For ▁a ▁point ▁, ▁the ▁expression ▁▁ ▁is ▁the ▁slope ▁of ▁the ▁sec ant ▁ ▁Because ▁both ▁the ▁functions ▁ ▁and ▁ ▁are ▁bij e ctions ▁from ▁ ▁to ▁, ▁and ▁ ▁a ▁bij ection ▁ ▁from ▁ ▁onto ▁ ▁, ▁where ▁ ▁is ▁the ▁slope ▁of ▁the ▁tang ent ▁at ▁ ▁, ▁for ▁▁ ▁we ▁get ▁ ▁( Rem ark : ▁For ▁ ▁we ▁have : ▁ ▁) ▁Hence ▁ ▁Because ▁the ▁s lop es ▁of ▁line ▁▁ ▁and ▁tang ent ▁ ▁both ▁are ▁, ▁it ▁follows ▁that ▁. ▁This ▁is ▁true ▁for ▁any ▁triangle ▁. ▁ ▁So : ▁( P 3 ) ▁of ▁the ▁ 3 - point ▁Pascal ▁theorem ▁holds ▁and ▁the ▁o val ▁is ▁a ▁non ▁deg ener ate