Geometry Problem
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Saturday, January 22, 2011
Problem 572: Right triangle, Median, Circle, Diameter, Excircle, Tangent
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Geometry Problem
Click the figure below to see the complete problem 572.
Zoom at: Geometry Problem 572
http://img827.imageshack.us/img827/3008/problem572.png
ReplyDeleteLet 2p= perimeter of triangle ABC and a, b, c are 3 sides of triangle ABC
Let O is the center of excircle with radius R and I is the center of the circle diameter BD and radius r.
We have R= p-BC=p-a =1/2(b+c-a) and r= 1/4AC=1/4a ( see attached picture)
OI^2=KI^2+OK^2=(R+1/4a)^2+(R-1/4c)^2
= 2R^2+1/2R(a-c)+1/16(a^2+c^2) = 2R^2+1/2R(a-c)+1/16b^2 (1)
(R+r)^2=(R+1/4b)^2=R^2+1/2R.b+1/16b^2 (2)
(1)- (2)= R^2+1/2R(a-c-b)= R*(R-1/2(b+c-a)) = 0 because R=1/2 (b+c-a)
So OI^2= (R+r)^2 and circle O will tangent to circle I
Peter Tran
Comment: Circle 1 is the nine-point circle which, by Feurbach's theorem, touches the in-circle and the three ex-circles, in particular Circle 2
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