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#1 2014-02-08 20:23:31

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

sort of power of a point

Let \overline{PA} and \overline{PC} be tangents from P to a circle. Let B and D be points on the circle such that B, D, and P are collinear. Prove that AB \cdot CD = BC \cdot DA.

This is driving me CRAZY and the moment. mad
several swearing smileys should make me feel better. swearswearswear


Good. You can read.

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#2 2014-02-08 22:53:50

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,586

Re: sort of power of a point

HI thedarktiger,

I know the feeling.  I still have some problems posted over a year ago on my 'to-do' list.  I remain optimistic about them.

EVEN LATER EDIT:

I thought this was going to come out easily; but it hasn't.  Nevertheless I think the approach is the right one ... I've just got in a muddle with my similar triangles.

Your problem is a little like a theorem about intersecting chords so I've used a similar method.  I haven't used the tangents at all; whoops!I haven't changed my mind about this.  If you pick any two points (not diametrically opposite) you can construct tangents that will intersect at P.  As P doesn't feature in the required equation, I cannot see why you cannot treat P as any point just to determine B and C.  So all the 'action' should be able to take place inside the circle.  As I type this, I'm already thinking I'm wrong so maybe I'll have a new approach soon smile )

I have now explored what happens if P isn't as specified and the property fails to hold.  Conclusion: what I wrote earlier is wrong so I shall delete it and go back to the drawing board.


Bob

View Image: thedarktiger8.gif

Last edited by bob bundy (2014-02-08 23:41:18)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2014-02-09 09:00:50

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,586

Re: sort of power of a point

hi,

It has driven me crazy too.  I have drawn and re-drawn the diagram with varied emphasis and called angles x, y and z and then found every other angle in terms of these.  I have experimented with similar triangles until my head feels like bursting.  And still no joy.  sad

With my geometry program I have verified that it is, indeed, true.  But I am at a loss as to how to prove it.

Very sorry.  Maybe tomorrow ..........

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2014-02-09 17:52:53

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,015

Re: sort of power of a point

Hi guys,

- look at your own responsibility smile

Last edited by anonimnystefy (2014-02-09 17:56:04)


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#5 2014-02-09 22:25:15

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

Re: sort of power of a point

Thanks for the anwser, anonimnystefy! I think my head has already burst. That problem was plain evil.
I'm sure anybody named bob will agree. aaaaaaggggghhhhhh.


Good. You can read.

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#6 2014-02-10 02:19:39

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,015

Re: sort of power of a point

You're welcome! smile

The problem is quite tricky!


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#7 2014-02-10 02:29:19

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,586

Re: sort of power of a point

hi Stefy,

Thanks for putting me out of my misery.  I'll be able to sleep now. sleep

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#8 2014-02-10 15:08:22

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,015

Re: sort of power of a point

You're welcome, but I sure am hoping you are using that as a figure of speech only. smile


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#9 2014-02-10 20:40:50

thedarktiger
Member
Registered: 2014-01-10
Posts: 91

Re: sort of power of a point

Yup. don't want a math problem to do THAT. big_smile


Good. You can read.

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#10 2015-07-02 03:26:18

RandomPieKevin
Member
Registered: 2015-07-02
Posts: 29

Re: sort of power of a point

I don't understand how anonimnystefy got the first two ratios.

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#11 2015-07-02 22:23:50

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,586

Re: sort of power of a point

hi RandomPieKevin

Welcome to the forum.

Angle BAP = ADP ( see http://www.mathisfunforum.com/viewtopic.php?id=17799 post 7 )

The ratio result then follows from "triangles ABP and DAP are similar".

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#12 2016-06-14 09:25:38

dazzle1230
Member
Registered: 2016-05-17
Posts: 45

Re: sort of power of a point

I still don't understand the property of the circle that makes angle BAP=ADP.
Is there another way to show that the triangles are similar?
I've tried using angle chasing, but I don't know which angles I should use to represent with the variables x and y.

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#13 2016-06-14 19:22:24

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,586

Re: sort of power of a point

hi dazzle1230

This comes from one of the angle properties of a circle.  Have a look at post 7 here:

http://www.mathisfunforum.com/viewtopic.php?id=17799

This relies on the other properties also proved in that thread.

I have also made a contents post for other geometry proofs which you will find here:

http://www.mathisfunforum.com/viewtopic.php?id=22506

Hope that helps,

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#14 2016-06-15 08:46:38

dazzle1230
Member
Registered: 2016-05-17
Posts: 45

Re: sort of power of a point

I see it now, thank you so much bob!

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#15 2016-06-15 20:05:41

thickhead
Member
Registered: 2016-04-16
Posts: 582

Re: sort of power of a point

Last edited by thickhead (2016-06-15 20:06:00)


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There is nothing impossible on earth.

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