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December 28th, 2017, 05:24 PM   #1
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Joined: Aug 2011
From: Nouakchott, Mauritania

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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 ! Attached Images MYMATHFORUM.jpg (9.6 KB, 23 views) December 29th, 2017, 08:29 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,927 Thanks: 2205 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. Thanks from Ould Youbba and johng40 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: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond Thanks from Ould Youbba December 29th, 2017, 06:11 PM   #4
Member

Joined: Aug 2011
From: Nouakchott, Mauritania

Posts: 85
Thanks: 14

Math Focus: Algebra, Cryptography
Good morning !
Quote:
 Originally Posted by skipjack 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.
I like this fast solution ! Well done ! December 29th, 2017, 06:16 PM #5 Member   Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography Salam ! Quote: Thanks for this useful information. I didn't expected that it is already a known theorem. Thanks from greg1313 Tags circle, interesting, question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Swami Mathsananda Algebra 1 August 4th, 2014 07:50 AM Saunak Algebra 2 September 6th, 2013 10:22 AM Bromster Number Theory 0 August 17th, 2013 11:02 AM galactus Calculus 4 January 13th, 2012 08:31 AM Saunak Real Analysis 1 December 31st, 1969 04:00 PM

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