Question Number 27422 by Rasheed.Sindhi last updated on 06/Jan/18
![A,B,C & D are four distinct points of a circle in such a way that chords AB & CD cut each other inside the circle at the point E. Consequently the circle is divided in four parts (AEC,CEB,BED & DEA). [AEC means the region outlined by AC^(⌢) ,AE^(−) & CE^(−) ] If AE : BE=a:b and CE : DE=c:d, what is ratio between the four parts of the circle?](Q27422.png)
$$\mathrm{A},\mathrm{B},\mathrm{C}\:\&\:\mathrm{D}\:\mathrm{are}\:\mathrm{four}\:\mathrm{distinct}\:\mathrm{points}\: \\ $$ $$\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{in}\:\mathrm{such}\:\mathrm{a}\:\mathrm{way}\:\mathrm{that}\:\mathrm{chords} \\ $$ $$\mathrm{AB}\:\&\:\mathrm{CD}\:\mathrm{cut}\:\mathrm{each}\:\mathrm{other}\:\mathrm{inside} \\ $$ $$\mathrm{the}\:\mathrm{circle}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{E}.\:\mathrm{Consequently} \\ $$ $$\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{divided}\:\mathrm{in}\:\mathrm{four}\:\mathrm{parts} \\ $$ $$\left(\mathrm{AEC},\mathrm{CEB},\mathrm{BED}\:\&\:\mathrm{DEA}\right). \\ $$ $$\left[\mathrm{AEC}\:\mathrm{means}\:\:\mathrm{the}\:\mathrm{region}\:\mathrm{outlined}\:\:\right. \\ $$ $$\left.\mathrm{by}\:\overset{\frown} {\mathrm{AC}},\overline {\mathrm{AE}}\:\&\:\overline {\mathrm{CE}}\right] \\ $$ $$\mathrm{If}\:\mathrm{AE}\::\:\mathrm{BE}=\mathrm{a}:\mathrm{b}\:\mathrm{and}\:\mathrm{CE}\::\:\mathrm{DE}=\mathrm{c}:\mathrm{d}, \\ $$ $$\mathrm{what}\:\mathrm{is}\:\mathrm{ratio}\:\mathrm{between}\:\mathrm{the}\:\mathrm{four} \\ $$ $$\mathrm{parts}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}? \\ $$
Commented byRasheed.Sindhi last updated on 07/Jan/18
Commented byRasheed.Sindhi last updated on 07/Jan/18
$$\mathrm{Is}\:\mathrm{the}\:\mathrm{data}\:\mathrm{sufficient}\:\mathrm{to}\:\mathrm{prove} \\ $$ $$\mathrm{the}\:\mathrm{result}?\:\mathrm{Perhaps}\:\mathrm{not}\:\mathrm{so}\:\mathrm{please} \\ $$ $$\mathrm{suppose}\:\mathrm{that}\:\angle\mathrm{AEC}=\theta \\ $$ $$ \\ $$