凸四边形外接椭圆存在性的初等证明

,,,,,:.,ABCD,,,.ABCD,°180,AC∠+∠180°,12,,BACDACαα∠∠3,BCAα∠4DCAα∠.(1)321,,ααα4α°90,ABCDCEFAC⊥,ABADEF.ABCDEFDCBA′′′′,AAC′∠,β,,β,DCBA′′′′.1tan,cos,ECACACACαβ′==1tantancosECEACACαβ′∠==′.2tantancosFACαβ′∠=,tan'EAF∠tantan1tantanEACFACEACFAC′′∠+∠=′′−∠∠21221tantancos)tan(tancosααβααβ−+=.,tanBCD′′∠43243tantancos)tan(tancosααβααβ−+=.DCBA′′′′,0)tan(=′′∠+′∠DCBFAE,0tan=′′∠+′∠DCBtgFAE,21221tantancos)tan(tancosααβααβ−+0tantancos)tan(tancos43243=−++ααβααβ,90,cos0,ββ≠°∴≠Qcosβ=123[(tantantanααα+234tantantanααα+3414121tantantantantantan)/(tanααααααα+1/2234tantantan)]ααα+++.,β,DCBA′′′′,,DCBA′′′′,,CBA,,D,ABCD,ABCD.,,°=′∠=′′∠90FAEDCB,EAC′∠+90FAC′∠=°,21tantancosααβ=,43tantancosααβ=,β.321,,ααα4α,4α,FDD,,′,,3costancot,tanBCDBCAβα′′′′∠=−∠=−12cos(tantanβαα=+1/22331tantantantan)αααα+,..,.BACDFE'B'A'D·22·