▁el lipse ▁that ▁is ▁tang ent ▁to ▁the ▁sides ▁of ▁a ▁triangle ▁ ▁at ▁its ▁mid points ▁ ▁is ▁called ▁the ▁Ste iner ▁in el lipse ▁of ▁triangle ▁. ▁ ▁Properties : ▁For ▁an ▁arbitrary ▁triangle ▁ ▁with ▁mid points ▁ ▁of ▁its ▁sides ▁the ▁following ▁statements ▁are ▁true : ▁a ) ▁There ▁exists ▁exactly ▁one ▁Ste iner ▁in el lipse . ▁b ) ▁The ▁center ▁of ▁the ▁Ste iner ▁in el lipse ▁is ▁the ▁cent roid ▁ ▁of ▁the ▁triangle ▁. ▁c 1 ) ▁The ▁triangle ▁ ▁has ▁the ▁same ▁cent roid ▁ ▁and ▁the ▁Ste iner ▁in el lipse ▁of ▁the ▁triangle ▁ ▁is ▁the ▁Ste iner ▁el lipse ▁of ▁the ▁triangle ▁. ▁ ▁c 2 ) ▁The ▁Ste iner ▁in el lipse ▁of ▁a ▁triangle ▁is ▁the ▁scale d ▁Ste iner ▁El lipse ▁with ▁scaling ▁factor ▁ 1 / 2 ▁and ▁the ▁cent roid ▁as ▁center . ▁Hence ▁both ▁elli ps es ▁have ▁the ▁same ▁ecc ent ric ity , ▁are ▁similar . ▁d ) ▁The ▁area ▁of ▁the ▁Ste iner ▁in el lipse ▁is ▁- times ▁the ▁area ▁of ▁the ▁triangle . ▁e ) ▁The ▁Ste iner ▁in el lipse ▁has ▁the ▁greatest ▁area ▁of ▁all ▁in elli ps es ▁of ▁the ▁triangle . ▁ ▁Proof ▁The ▁proofs ▁of ▁properties ▁a ), b ), c ) ▁are ▁based ▁on ▁the ▁following ▁properties ▁of ▁an ▁aff ine ▁mapping : ▁ 1 ) ▁any ▁triangle ▁can ▁be ▁considered ▁as ▁an ▁aff ine ▁image ▁of ▁an ▁equ il ater al ▁triangle .