▁ 2 ) ▁Mid points ▁of ▁sides ▁are ▁mapped ▁onto ▁mid points ▁and ▁cent roid s ▁on ▁cent roid s . ▁The ▁center ▁of ▁an ▁el lipse ▁is ▁mapped ▁onto ▁the ▁center ▁of ▁its ▁image . ▁Hence ▁its ▁suff ice ▁to ▁prove ▁properties ▁a ), b ), c ) ▁for ▁an ▁equ il ater al ▁triangle : ▁a ) ▁To ▁any ▁equ il ater al ▁triangle ▁there ▁exists ▁an ▁inc irc le . ▁It ▁touch es ▁the ▁sides ▁at ▁its ▁mid points . ▁There ▁is ▁no ▁other ▁( non - de generate ) ▁con ic ▁section ▁with ▁the ▁same ▁properties , ▁because ▁a ▁con ic ▁section ▁is ▁determined ▁by ▁ 5 ▁points / t ang ents . ▁b ) ▁By ▁a ▁simple ▁calculation . ▁c ) ▁The ▁circum circle ▁is ▁mapped ▁by ▁a ▁scaling , ▁with ▁factor ▁ 1 / 2 ▁and ▁the ▁cent roid ▁as ▁center , ▁onto ▁the ▁inc irc le . ▁The ▁ecc ent ric ity ▁is ▁an ▁invariant . ▁d ) ▁The ▁ratio ▁of ▁areas ▁is ▁invariant ▁to ▁aff ine ▁transformations . ▁So ▁the ▁ratio ▁can ▁be ▁calculated ▁for ▁the ▁equ il ater al ▁triangle . ▁e ) ▁See ▁In el lipse . ▁ ▁Param et ric ▁representation ▁and ▁semi - ax es ▁ ▁Param et ric ▁representation : ▁ ▁Because ▁a ▁Ste iner ▁in el lipse ▁of ▁a ▁triangle ▁ ▁is ▁a ▁scale d ▁Ste iner ▁el lipse ▁( factor ▁ 1 / 2 , ▁center ▁is ▁cent roid ) ▁one ▁gets ▁a ▁paramet ric ▁representation ▁derived ▁from