Math Is Fun Forum

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.
several swearing smileys should make me feel better.

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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 )

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

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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. 

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: sort of power of a point

Hi guys,

- look at your own responsibility

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

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“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.

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.

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Good. You can read.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: sort of power of a point

You're welcome!

The problem is quite tricky!

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“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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: sort of power of a point

hi Stefy,

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

Bob

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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.

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“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.

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.

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Good. You can read.

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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: sort of power of a point

I see it now, thank you so much bob!

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: sort of power of a point

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

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