▁result ▁is ▁one ▁of ▁the ▁deeper ▁results ▁in ▁ax iom atic ▁plane ▁geometry . ▁It ▁is ▁often ▁used ▁in ▁proofs ▁to ▁justify ▁the ▁statement ▁that ▁a ▁line ▁through ▁a ▁vertex ▁of ▁a ▁triangle ▁lying ▁inside ▁the ▁triangle ▁meets ▁the ▁side ▁of ▁the ▁triangle ▁opposite ▁that ▁vertex . ▁This ▁property ▁was ▁often ▁used ▁by ▁Eu cl id ▁in ▁his ▁proofs ▁without ▁explicit ▁just ification . ▁ ▁Some ▁modern ▁treat ments ▁( not ▁Eu cl id ' s ) ▁of ▁the ▁proof ▁of ▁the ▁theorem ▁that ▁the ▁base ▁angles ▁of ▁an ▁is os cel es ▁triangle ▁are ▁con gru ent ▁start ▁like ▁this : ▁Let ▁ABC ▁be ▁a ▁triangle ▁with ▁side ▁AB ▁con gru ent ▁to ▁side ▁AC . ▁Draw ▁the ▁angle ▁bis ector ▁of ▁angle ▁A ▁and ▁let ▁D ▁be ▁the ▁point ▁at ▁which ▁it ▁meets ▁side ▁BC . ▁And ▁so ▁on . ▁The ▁just ification ▁for ▁the ▁existence ▁of ▁point ▁D ▁is ▁the ▁often ▁un st ated ▁cross bar ▁theorem . ▁For ▁this ▁particular ▁result , ▁other ▁proofs ▁exist ▁which ▁do ▁not ▁require ▁the ▁use ▁of ▁the ▁cross bar ▁theorem . ▁ ▁See ▁also ▁ ▁Found ations ▁of ▁geometry ▁ ▁Jordan ▁curve ▁theorem ▁ ▁Notes ▁ ▁References ▁▁▁▁▁▁▁▁▁ ▁Category : E uclidean ▁plane ▁geometry ▁Category : The or ems ▁in ▁plane ▁geometry ▁Category : Found ations ▁of ▁geometry <0x0A> </s> ▁J ut land ▁cattle ▁are ▁a ▁rare ▁Dan ish ▁bre ed ▁of ▁cattle ▁used ▁in ▁both ▁da iry ▁and ▁be ef ▁production . ▁B red ▁from ▁the ▁ind igen ous ▁cattle ▁of ▁J