Leg end rian ▁geometry ▁ ▁In ▁the ▁same ▁paper , ▁De hn ▁also ▁constructed ▁an ▁example ▁of ▁a ▁non - Leg end rian ▁geometry ▁where ▁there ▁are ▁infinitely ▁many ▁lines ▁through ▁a ▁point ▁not ▁meeting ▁another ▁line , ▁but ▁the ▁sum ▁of ▁the ▁angles ▁in ▁a ▁triangle ▁exceed s ▁ π . ▁Riemann ' s ▁elli ptic ▁geometry ▁over ▁ Ω ( t ) ▁consists ▁of ▁the ▁project ive ▁plane ▁over ▁ Ω ( t ), ▁which ▁can ▁be ▁identified ▁with ▁the ▁aff ine ▁plane ▁of ▁points ▁( x : y : 1 ) ▁together ▁with ▁the ▁" line ▁at ▁infinity ", ▁and ▁has ▁the ▁property ▁that ▁the ▁sum ▁of ▁the ▁angles ▁of ▁any ▁triangle ▁is ▁greater ▁than ▁ π ▁The ▁non - Leg end rian ▁geometry ▁consists ▁of ▁the ▁points ▁ ▁( x : y : 1 ) ▁of ▁this ▁aff ine ▁sub space ▁such ▁that ▁tx ▁and ▁ty ▁are ▁finite ▁( where ▁as ▁above ▁t ▁is ▁the ▁element ▁of ▁ Ω ( t ) ▁represented ▁by ▁the ▁identity ▁function ). ▁Leg endre ' s ▁theorem ▁states ▁that ▁the ▁sum ▁of ▁the ▁angles ▁of ▁a ▁triangle ▁is ▁at ▁most ▁ π , ▁but ▁assumes ▁Arch im edes ' s ▁ax iom , ▁and ▁De hn ' s ▁example ▁shows ▁that ▁Leg endre ' s ▁theorem ▁need ▁not ▁hold ▁if ▁Arch im edes ' ▁ax iom ▁is ▁dropped . ▁ ▁References ▁ ▁Category : Ge ometry ▁Category : Non - E uclidean ▁geometry <0x0A> </s> ▁Mo een ▁Mun ir ▁Ali ▁( born ▁