Let-S-be-a-circle-with-centre-O-A-chord-AB-not-a-diameter-divides-S-into-two-regions-R-1-and-R-2-such-that-O-belongs-to-R-2-Let-S-1-be-a-circle-with-centre-in-R-1-touching-AB-at-X-and-S-inte

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Question Number 19699 by Tinkutara last updated on 14/Aug/17

$$\mathrm{Let}S\mathrm{be}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{centre}O.\mathrm{A}\mathrm{chord}$$$$AB,\mathrm{not}\mathrm{a}\mathrm{diameter},\mathrm{divides}S\mathrm{into}\mathrm{two}$$$$\mathrm{regions}{R}_1\mathrm{and}{R}_2\mathrm{such}\mathrm{that}O\mathrm{belongs}$$$$\mathrm{to}{R}_2.\mathrm{Let}{S}_1\mathrm{be}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{centre}\mathrm{in}$$$$R_1,\mathrm{touching}AB\mathrm{at}X\mathrm{and}S\mathrm{internally}.$$$$\mathrm{Let}{S}_2\mathrm{be}\mathrm{a}\mathrm{circle}\mathrm{with}\mathrm{centre}\mathrm{in}{R}_2,$$$$\mathrm{touching}AB\mathrm{at}Y,\mathrm{the}\mathrm{circle}S\mathrm{internally}$$$$\mathrm{and}\mathrm{passing}\mathrm{through}\mathrm{the}\mathrm{centre}\mathrm{of}S.$$$$\mathrm{The}\mathrm{point}X\mathrm{lies}\mathrm{on}\mathrm{the}\mathrm{diameter}$$$$\mathrm{passing}\mathrm{through}\mathrm{the}\mathrm{centre}\mathrm{of}{S}_2\mathrm{and}$$$$\mathrm{\angle }YXO=30°.\mathrm{If}\mathrm{the}\mathrm{radius}\mathrm{of}{S}_2\mathrm{is}100$$$$\mathrm{then}\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{radius}\mathrm{of}{S}_1?$$

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