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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

4 secants circle geometry

Let ABCD be a cyclic quadrilateral. Let P be the intersection of \overline{AD} and \overline{BC}, and let Q be the intersection of \overline{AB} and \overline{CD}. Prove that the angle bisectors of \angle DPC and \angle AQD are perpendicular.

I just don't know how to get this one. This problem in one word=aggggggghhhhhh.
would power of a point help?
thanks!

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: 4 secants circle geometry

Hi;

Hi; I put the image in for you. Also you need to use the math tags for your latex.

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: 4 secants circle geometry

hi thedarktiger,

This time I'm more confident about what to do.  (But I haven't fully tried it yet.  )

Call angle APB = x and ABP = y

So DAB = x + y  => DCB = 180 - (x + y)  => BCQ = x + y

So get an expression in terms of x and y for BQC.

Let the bisectors cross at R, and let RQ cross BC at S

Get an expression in terms of x and y for RPB and RQB, then RSB.

Hence for PRQ.

EDIT:  Thought I'd better check it ..... it does work. 

Bob

Last edited by Bob (2014-01-28 00:02:42)

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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

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,876

Re: 4 secants circle geometry

Hi,

Drew it up in Geogebra, which gave the two bisectors as being perpendicular to each other.

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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: 4 secants circle geometry

wow! thanks! @bob bundy, you are really good at geometry. phrostister and bobbym, thanks for the pics!

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phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,876

Re: 4 secants circle geometry

Here is a little video of post #4, demonstrating the perpendicular property of the two bisectors in various cyclic quadrilateral shapes. Done in Geogebra by randomly dragging the points of the quadrilateral along the curve of the circle.

Last edited by phrontister (2014-01-29 23:44:45)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: 4 secants circle geometry

thanks a lot for the video! So I guess they are always perpendicular.
I think there should be some kind of theorem ... maybe the bob-tister-bob theorem of secants.
Maybe I will make a wikipedia page.

Last edited by thedarktiger (2014-02-02 21:56:00)

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championmathgirl

Member

Registered: 2015-06-01

Posts: 20

Re: 4 secants circle geometry

@bob bundy, I was wondering how you got angle DCB = 180 - (x + y) => BCQ = x + y.

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Girls can be just as good as boys at math. We just need to get the same encouragement.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: 4 secants circle geometry

hi championmathgirl

Welcome to the forum.

There is always one circle that goes through the three points of any triangle.  A 4th point may or may not be on this circle too.  If it is the quadrilateral is called cyclic.  ABCD is such a quadrilateral.  All cyclic quadrilaterals have the property that opposite angles add up to 180 degrees.  I proved this here:

http://www.mathisfunforum.com/viewtopic.php?id=17799  starting at post 4.

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

Wangotango

Member

Registered: 2016-07-29

Posts: 6

Re: 4 secants circle geometry

Can someone help me find RQB, RSB. And also can someone tell me how that relates to PQR?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: 4 secants circle geometry

Last edited by thickhead (2016-07-29 19:31:34)

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Wangotango

Member

Registered: 2016-07-29

Posts: 6

Re: 4 secants circle geometry

Can you prove that the exterior angle and remote interior angle of a cylic quadrilateral is equal?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: 4 secants circle geometry

It depends on what is the starting point. If you start with the sum of opposite angles is

it is one step. Else you have to start with the angle inscribed in an arc is half the measure of arc and show that the sum of opposite angles is

first.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Wangotango

Member

Registered: 2016-07-29

Posts: 6

Re: 4 secants circle geometry

Ok thx I proved it, but how does RSB=RQB+D help? I'm still not sure how to finish this proof...

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: 4 secants circle geometry

What exactly you need to prove? I thought you meant angle PDQ.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Wangotango

Member

Registered: 2016-07-29

Posts: 6

Re: 4 secants circle geometry

We're trying to prove that where the angle bisectors intersect is perpendicular

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: 4 secants circle geometry

I thought it was proved by phrontister. Any way I have have given the proof  in some recent thread. My current response was just for #10.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: 4 secants circle geometry

Hi Wangotango,
Please see the following thread
http://www.mathisfunforum.com/viewtopic.php?id=23185

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}