See complete Problem 160
Triangle, Incircle, Incenter, Circumcircle, Circumcenter, Inradius. Level: High School, SAT Prep, College geometry
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Monday, August 11, 2008
Elearn Geometry Problem 160
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d=R-(r-OI)=R-r+OI,e=R-r-OI.
ReplyDeleteSo, de=(R-r)²-OI²=(R²-2Rr-OI²)+r².
But, OI²=R²-2Rr (Problem 155)
∴de=r²
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ReplyDeleteJoin B with I , then extend BI to meet the circle at point P.
ReplyDeleteJoin D with P and B with E.
Triangles DPI and BEI are similar therefore we get :
BI/IE=DI/IP
(BI)(IP)=(DI)(IE)
We have 2R=d+e+2r and from problem 154 we have:
(BI)(IP) =2Rr
(BI)(IP) =(d+e+2r)r
and
DI= d+r
IE=e+r
therefore
2rR=(e+r)(d+r)
2r^2+r(d+e)=r^2+r(d+e)+de
r^2=de