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Geometry Problem. Post your solution in the comments box below.Level: Mathematics Education, High School, Honors Geometry, College.Click the diagram below for more details.
Let the radius be R and let OE = sLet AG = a. GF = b and GH = cFrom similar Tr.s(R+s)(R-s) = CE^2 = ED^2. (1)(R+s)(2R) = a(a+b). (2)(R+s)^2 = a(a-c). (3)(2) - (3)(R+s)(R-s) = a(a+b) - a(a-c) = a(b+c) which from (1) equals CE ^2 which is the required result. Sumith PeirisMoratuwaSri Lanka
Since BEGF concyclic, thusCE² = AE×EB= AE×AB - AE²= AG×AF - AG×AH= AG×HF
For (1) I've used similar Tr.s AED & BED, for (2) Tr.s ABF and AEG and for (3) Tr.s AEH & AEG