A-B-C-amp-D-are-four-distinct-points-of-a-circle-in-such-a-way-that-chords-AB-amp-CD-cut-each-other-inside-the-circle-at-the-point-E-Consequently-the-circle-is-divided-in-four-parts-AEC-CEB-BED

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Question Number 27422 by Rasheed.Sindhi last updated on 06/Jan/18

$$\mathrm{A},\mathrm{B},\mathrm{C}\mathrm{\&}\mathrm{D}\mathrm{are}\mathrm{four}\mathrm{distinct}\mathrm{points}$$$$\mathrm{of}\mathrm{a}\mathrm{circle}\mathrm{in}\mathrm{such}\mathrm{a}\mathrm{way}\mathrm{that}\mathrm{chords}$$$$\mathrm{AB}\mathrm{\&}\mathrm{CD}\mathrm{cut}\mathrm{each}\mathrm{other}\mathrm{inside}$$$$\mathrm{the}\mathrm{circle}\mathrm{at}\mathrm{the}\mathrm{point}\mathrm{E}.\mathrm{Consequently}$$$$\mathrm{the}\mathrm{circle}\mathrm{is}\mathrm{divided}\mathrm{in}\mathrm{four}\mathrm{parts}$$$$(\mathrm{AEC},\mathrm{CEB},\mathrm{BED}\mathrm{\&}\mathrm{DEA}).$$$$[\mathrm{AEC}\mathrm{means}\mathrm{the}\mathrm{region}\mathrm{outlined}$$$$\mathrm{by}\stackrel{⌢}{\mathrm{AC}},\overline{\mathrm{AE}}\mathrm{\&}\overline{\mathrm{CE}}]$$$$\mathrm{If}\mathrm{AE}:\mathrm{BE}=\mathrm{a}:\mathrm{b}\mathrm{and}\mathrm{CE}:\mathrm{DE}=\mathrm{c}:\mathrm{d},$$$$\mathrm{what}\mathrm{is}\mathrm{ratio}\mathrm{between}\mathrm{the}\mathrm{four}$$$$\mathrm{parts}\mathrm{of}\mathrm{the}\mathrm{circle}?$$

Commented by Rasheed.Sindhi last updated on 07/Jan/18

Commented by Rasheed.Sindhi last updated on 07/Jan/18

$$\mathrm{Is}\mathrm{the}\mathrm{data}\mathrm{sufficient}\mathrm{to}\mathrm{prove}$$$$\mathrm{the}\mathrm{result}?\mathrm{Perhaps}\mathrm{not}\mathrm{so}\mathrm{please}$$$$\mathrm{suppose}\mathrm{that}\mathrm{\angle }\mathrm{AEC}=\theta$$$$$$

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