▁the ▁trig on ometric ▁representation ▁of ▁the ▁Ste iner ▁el lipse ▁: ▁▁ ▁The ▁ 4 ▁vertices ▁of ▁the ▁Ste iner ▁in el lipse ▁are ▁ ▁where ▁ ▁is ▁the ▁solution ▁of ▁ ▁with ▁▁ ▁Sem i - ax es : ▁ ▁With ▁the ▁abbre vi ations ▁ ▁one ▁gets ▁for ▁the ▁semi - ax es ▁: ▁▁ ▁The ▁linear ▁ecc ent ric ity ▁ ▁of ▁the ▁Ste iner ▁in el lipse ▁is ▁ ▁Tr il ine ar ▁equation ▁ ▁The ▁equation ▁of ▁the ▁Ste iner ▁in el lipse ▁in ▁tr il ine ar ▁coordinates ▁for ▁a ▁triangle ▁with ▁side ▁lengths ▁a , ▁b , ▁c ▁( with ▁these ▁parameters ▁having ▁a ▁different ▁meaning ▁than ▁previously ) ▁is ▁ ▁where ▁x ▁is ▁an ▁arbitrary ▁positive ▁constant ▁times ▁the ▁distance ▁of ▁a ▁point ▁from ▁the ▁side ▁of ▁length ▁a , ▁and ▁similarly ▁for ▁b ▁and ▁c ▁with ▁the ▁same ▁multip lic ative ▁constant . ▁ ▁Other ▁properties ▁ ▁The ▁lengths ▁of ▁the ▁semi - major ▁and ▁semi - min or ▁axes ▁for ▁a ▁triangle ▁with ▁sides ▁a , ▁b , ▁c ▁ ▁are ▁ ▁where ▁ ▁According ▁to ▁M arden ' s ▁theorem , ▁if ▁the ▁three ▁vertices ▁of ▁the ▁triangle ▁are ▁the ▁complex ▁zeros ▁of ▁a ▁cub ic ▁polynomial , ▁then ▁the ▁fo ci ▁of ▁the ▁Ste iner ▁in el lipse ▁are ▁the ▁zeros ▁of ▁the ▁derivative ▁of ▁the ▁polynomial . ▁ ▁The ▁major ▁axis ▁of ▁the ▁Ste iner ▁in el lipse ▁is ▁the ▁line ▁of ▁best ▁orthogonal ▁fit ▁for ▁the ▁vertices .