AN IMO PROBLEM
On the circle $k$ there points $A,B,C$ are given.construct the fourth point on the circle $D$ such that one can inscribe a circle in $ABCD$.
Last edited by Moon on Sat Dec 11, 2010 8:58 pm, edited 1 time in total.
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- Raiyan Jamil
- Posts:138
- Joined:Fri Mar 29, 2013 3:49 pm
Re: AN IMO PROBLEM
Firstly, in the case $AB=BC$, the point $D$ is just the antipode of $B$.So let $AB\neq BC$
1)We draw point $E$ extending $AB$ past $B$ such that $BE=BC$.
2)Let $d=|AB-BC|$.We draw a circle with radius d and centre $C$.
3)We draw the circle $ACE$.
4)Let the two circles intersect at $X,Y$ such that $X$ lies on the side of $AC$ not containing $B$.
5)Let $CX$ intersect circle $ABC$ at $D$. This $D$ is indeed the required point which can be proved by proving $AB+CD=AD+BC$.
Note:Proofing that this is the required point isn't hard [left for the reader] but solving this includes the proof in an official competition.
1)We draw point $E$ extending $AB$ past $B$ such that $BE=BC$.
2)Let $d=|AB-BC|$.We draw a circle with radius d and centre $C$.
3)We draw the circle $ACE$.
4)Let the two circles intersect at $X,Y$ such that $X$ lies on the side of $AC$ not containing $B$.
5)Let $CX$ intersect circle $ABC$ at $D$. This $D$ is indeed the required point which can be proved by proving $AB+CD=AD+BC$.
Note:Proofing that this is the required point isn't hard [left for the reader] but solving this includes the proof in an official competition.
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