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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Intersecting circles

Circles S and T have radii 1, and intersect at A and B. The distance between their centers is \sqrt{2}.

Let P be a point on major arc AB of circle S, and let \overline{PA} and \overline{PB} intersect circle T again at C and D, respectively. Show that \overline{CD} is a diameter of circle T.

this has me stumped. I found that minor arc AB = arc CD/2 (CD opposite AB) but this doesent help much
thanks a lot I really want to see how to do this one!

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,136

Re: Intersecting circles

hi thedarktiger

In my diagram, I've called the centres S and T.

I think this hinges on the fact ST = root 2

We already know that AT and BT are 1, so by the inverse of Pythagoras, ATB = 90

If you call angle CAT = x and TBD = y, you can get all the other angles in the diagram in terms of x and y.  (Don't forget for example that triangle CAT is isosceles.)

The aim is to show that angle CAD = 90, because then CD is a diameter by the 'angle subtended by an arc at the edge is half that at the centre' theorem. 

LATER EDIT:  Hmmm.  That should have worked but I'm still chasing angles round the diagram at the moment.  I'll post again when I get there. 

Bob

Last edited by Bob (2014-01-26 23:37:51)

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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: 10,136

Re: Intersecting circles

hi again.

Done!  But not like I said, so I'll start again.

Same diagram and x and y as before.

Join AB

As ASB = 90 => APB = 45

Find PAT and PBT in terms of x and y and use the angle sum of the quadrilateral PATB to get an expression for x + y.

(Note.  The property continues to hold as P is moved about and x and y change individually, but their sum stays constant.)

Now calculate angle CTD in terms of x and y and using the above expression eliminate both x and y.

If you're left with 180, then CD is a diameter.

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

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: Intersecting circles

Thanks bob bundy! Sorry I couldn't reply right away, because my family moved to Australia
a month ago and the time difference is 8 hours I was sleeping.
Just one thing... could you explain why ASB = 90? It seems the angle would change as the circles came closer, right?

Oh I got it because they are both radii and ab=sqrt(2) they have to be perpendicular.
thank you bob Ive got it!
by the way, check out the oatmeal comics. He has some very funny comics on the Bob cats.

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