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Find the Ratio (
Posted on 20070820
)
I (radius=r) is the incircle of triangle ABC (right angled at B) and O (radius=R) is another circle which touches the extended side BC and AC. Find the ratio (R/r) if the point of intersection of circles I and O is midpoint of AC.
Submitted by
Praneeth
Rating:
3.2500
(4 votes)
Solution:
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3+2*√2
The tangents from an external point to the circle are equal. As Incircle touches hypotenuse at midpoint, ABC is isosceles right angledtriangle.
=> BC=AC/√2
Draw perpendiculars from I and O on to BC and points of intersection beP and Q respectively. Draw perpendicular from I to OQ to meet at R.
http://perplexus.info/show.php?pid=5713&cid=38269 Now BC=r+PC, AC=2*PC.
=> (r+PC)*√2=2*PC => PC=(√2 +1)*r (1)
Consider triangle IOR
IR=2*PC
IO=r+R
OR=Rr(Assume R>r)
As it is aright angled triangle,
4*PC˛+(Rr)˛=(R+r)˛
=> PC = √(R*r)
Substitute this in eq(1)
√(R*r)=(√2 +1)*r
=> √(R/r)=√2 +1
=> R/r = 3+2*√2.
Other solutions are given in comments by
Bractals
and
Charlie
.
Comments: (
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Subject
Author
Date
Answer
K Sengupta
20080810 06:34:25
solution
Charlie
20070820 11:48:05
Solution
Bractals
20070820 11:23:46
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