## Wednesday, December 28, 2011

### Problem 708: Circle, Tangent, Intersecting, Lune, Area, Diameter

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 708.

1. let R= radius of circle F, diameter AB
r=radius of circle G, diameter CD
Denote Area(X,Y)= area of circle center X, radius Y
Note that Triangle AFO congrurence to tri. OGC (case SAS)
so OA^2=OC^2= R^2+r^2
Blue area= Area(O,OA)- white area
Yellow area= Area(F,R)+ Area(G,r)-White area
But Area(O,OA)=pi*(R^2+r^2)=Area(F,R)+Area(G,r)
so Blue area=white Area
Peter Tran

2. 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^2
x=r
Peter Tran

3. 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 = 0
2(R²–r²)–2d(R+r) = 0
d=R–r,R=r+d
W+B=Π[r²+(r+d)²]=Π(R²+r²)=W+Y
Hence B=Y

4. Please refer to my solution for Problem 708:
d denotes distance OE.
Pravin

5. 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!!!

6. To Κλεάνθης Ξενιτίδης
see picture in the following link:
http://img88.imageshack.us/img88/8311/problem708.png
FO=r per my clarification above and FG=R+r so OG=R
triangle FAO congruence to tri. GOC ( case SAS)