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December 28th, 2017, 05:24 PM  #1 
Member Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography  Interesting Circle Question
Salam ! This is your Christmas gift from me . Let $\displaystyle O$ be the center of a circle. $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ and $\displaystyle D$ are points of this circle such that the straight lines $\displaystyle (AC)$ and $\displaystyle (BD)$ are perpendicular and secant in $\displaystyle I$. Let $\displaystyle H$ be the midpoint of the segment $\displaystyle [BC]$. In the triangle $\displaystyle IAD$, let $\displaystyle J$ be the foot of the altitude from $\displaystyle I$. Show that the points $\displaystyle I$, $\displaystyle J$ and $\displaystyle H$ are aligned (collinear). Good luck ! 
December 29th, 2017, 08:29 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
As HB = HI, $\small\angle$HIB = $\small\angle$HBI = $\small\angle$CAD = $\small\angle$DIJ, and so the points H, I and J are collinear.
Last edited by skipjack; December 29th, 2017 at 09:19 AM. 
December 29th, 2017, 08:54 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,814 Thanks: 1046 Math Focus: Elementary mathematics and beyond  
December 29th, 2017, 06:11 PM  #4 
Member Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography  
December 29th, 2017, 06:16 PM  #5  
Member Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography 
Salam ! Quote:  

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