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Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the complete problem 708.
let R= radius of circle F, diameter AB r=radius of circle G, diameter CDDenote Area(X,Y)= area of circle center X, radius YNote that Triangle AFO congrurence to tri. OGC (case SAS)so OA^2=OC^2= R^2+r^2Blue area= Area(O,OA)- white areaYellow area= Area(F,R)+ Area(G,r)-White areaBut Area(O,OA)=pi*(R^2+r^2)=Area(F,R)+Area(G,r)so Blue area=white AreaPeter Tran
Clarifications of above solution from Peter Tran:Position of O on segment FG is determined by equation:OA^2=R^2+r^2=OC^2=(R+r-x)^2+r^2x=rPeter Tran
Let W,B,Y denote the areas shaded white, blue and yellow respectively.W+B=Π.OA²=Π[R²+(R–d)²]W+B=Π.OC²=Π[r²+(r + d)²]∴R²–r²+(R–d)²–(r+d)² = 02(R²–r²)–2d(R+r) = 0d=R–r,R=r+dW+B=Π[r²+(r+d)²]=Π(R²+r²)=W+YHence B=Y
Please refer to my solution for Problem 708: d denotes distance OE.Pravin
Peter, that is the way i was trying to solve the problem, but i couldn't prove that the two triangles are congruent. And i didn't i understand how you proved this Congruence.Pravin's solution seems to me correct.Once you prove that d=R-r you can easily prove the Congruence of the triangles, but the proof you gave is more direct, so we don't need to prove the Congruence finally.P.S. We can see that angles AOC and AEC are right. (if we move the point O on the segment FG we will not find any other right angle.)sorry for my english!!!
To Κλεάνθης Ξενιτίδης see picture in the following link:http://img88.imageshack.us/img88/8311/problem708.pngFO=r per my clarification above and FG=R+r so OG=Rtriangle FAO congruence to tri. GOC ( case SAS)