# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Fri, 25 Jul 2014 04:37:56 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 9 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Why can't I get this...?Posted: Fri, 4 May 2012 23:19:01 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
Given a rectangle , construct circles such with centers with radii respectively, such that: , ( is the length of the diagonal of rectangle ). Construct the common external tangents of circles and . Construct the common external tangents of circles and . These lines form a quadrilateral. Show that one can inscribe a circle into this quadrilateral.

My Work:
Well the only thing that I know about circles being inscribed in quadrilaterals, is that we can inscribe a circle in a quadrilateral iff:
,

I have no idea how to use the fact that

At first I thought, "hmmmm the condition looks suspiciously like ,"

So, I thought similar triangles should destroy this problem. But, I couldn't find any similar triangles that could relate to the sides of the quadrilateral that we construct.

I feel like I'm being really dumb... >_<

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Why can't I get this...?Posted: Sat, 5 May 2012 08:16:05 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6777
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
rdj5933mile5math64 wrote:
Given a rectangle , construct circles such with centers with radii respectively, such that: , ( is the length of the diagonal of rectangle ). Construct the common external tangents of circles and . Construct the common external tangents of circles and . These lines form a quadrilateral. Show that one can inscribe a circle into this quadrilateral.

My Work:
Well the only thing that I know about circles being inscribed in quadrilaterals, is that we can inscribe a circle in a quadrilateral iff:
,

I have no idea how to use the fact that

At first I thought, "hmmmm the condition looks suspiciously like ,"

So, I thought similar triangles should destroy this problem. But, I couldn't find any similar triangles that could relate to the sides of the quadrilateral that we construct.

I feel like I'm being really dumb... >_<

Maybe you can try to prove directly that the centre of the rectangle is equidistance from all four sides of the quadrilateral, which follows trivially from ...

Then prove that indeed the feet of the perpendiculars lie inside those four sides.

_________________

Top

 Post subject: Re: Why can't I get this...?Posted: Sat, 5 May 2012 17:34:28 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
outermeasure wrote:
Maybe you can try to prove directly that the centre of the rectangle is equidistance from all four sides of the quadrilateral, which follows trivially from ...

outermeasure wrote:
Then prove that indeed the feet of the perpendiculars lie inside those four sides.

Darn which is also trivial. It's a little tricky to formalize but very intuitive.

Thank You outermeasure!

EDIT (a question): I take it that it was a good diagram that provided the intuition for your solution. Do you know where I might be able to find a good compass? Also, how do I get quicker at drawing good diagrams? I know basic Euclidean construction stuff, but it still takes me a long time to construct diagrams.

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Why can't I get this...?Posted: Sat, 5 May 2012 18:45:32 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 13970
Location: Austin, TX
Pictures are like any other skill, earned by hours of practice and repetition.

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: Why can't I get this...?Posted: Sun, 6 May 2012 00:36:44 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
Pictures are like any other skill, earned by hours of practice and repetition.

Thank you for the words of wisdom!

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Why can't I get this...?Posted: Sun, 6 May 2012 18:30:34 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6777
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
rdj5933mile5math64 wrote:
I take it that it was a good diagram that provided the intuition for your solution. Do you know where I might be able to find a good compass? Also, how do I get quicker at drawing good diagrams? I know basic Euclidean construction stuff, but it still takes me a long time to construct diagrams.

I don't think IMO allows compasses. Of course, you can use your glass to draw a circle in IMO

Actually I was looking at ways to use the condition which points me to the diagonals and the centre of the rectangle, as they are the only apparent place to look where the equality manifests itself.

_________________

Top

 Post subject: Re: Why can't I get this...?Posted: Sun, 6 May 2012 20:34:16 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
outermeasure wrote:
I don't think IMO allows compasses. Of course, you can use your glass to draw a circle in IMO

Lol IMO allowed them last year. If you don't mind me asking, have you done the IMO before?

_________________
math puns are the first sine of madness
-JDR

Top

 Post subject: Re: Why can't I get this...?Posted: Mon, 7 May 2012 10:28:42 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6777
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
rdj5933mile5math64 wrote:
outermeasure wrote:
I don't think IMO allows compasses. Of course, you can use your glass to draw a circle in IMO

Lol IMO allowed them last year. If you don't mind me asking, have you done the IMO before?

No, I am not a former IMO contestant.

And I'm pretty sure that the IMO didn't allow compasses a decade ago (I heard the idea of using the glass of water to draw circles from a friend of mine who did IMO back then), although national olympiads always do. They must have changed the regulation a few years ago.

_________________

Top

 Post subject: Re: Why can't I get this...?Posted: Mon, 7 May 2012 13:49:36 UTC
 S.O.S. Oldtimer

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 145
Location: Hockeytown
outermeasure wrote:
No, I am not a former IMO contestant.

Neither am I - and I don't think that I ever will be unless my parents decide to move to Guam or Puerto Rico lol (I have terrible luck with AIMEs - last year (first time I heard of the AIME exam) one of the questions was misworded which cost me an hour and this year I made 4 dumb mistakes.).

outermeasure wrote:
Actually I was looking at ways to use the condition which points me to the diagonals and the centre of the rectangle, as they are the only apparent place to look where the equality manifests itself.

Thanks for the intuition!

_________________
math puns are the first sine of madness
-JDR

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 9 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous