Circle Geo (1 Viewer)

[Xayma]

Originally posted by Faera
Well, now I know what circumscribes truly means- and considering i wasted hours of my life on this, I wont be forgetting it any time soon. -.-'
Thanks, guys.

Actually it is used incorrectly in this case, circumscribe means to enclose it in a circle (ie the circle should have surronded it)

I dont understand the first one, hmm I need a diagram, the way Im reading it is that T, P and Q would be points on the outside but it wouldnt be that because then they would obviously touch at T

 

[Faera]

err...
ive got a diagram made up
but dont know how to attach..
0.o

 

[Xayma]

Above the box where you can post a reply is a button called post reply it looks like this

Click on it and then you will see attachements and go browse and add it that way.

 

[Faera]

ahh
genius.

=)
thanks!

 

[Grey Council]

To prove they touch over there, you hafta prove there is a common tangent at T. Hrm.
This IS insane. lol, if i start to draw anymore lines, my diagram is gonna get messed up quicksmart.
hrm

 

[Xayma]

Thats what a CAD program is for , plus my circles are actually circles

 

[Grey Council]

lol, you can't seriously expect her to get a CAD program just to draw maths. >.<

anyway, any ideas Xayma? On how to do this one.

 

[Xayma]

I have a demo of one it works well.

Well if we want to prove it is a common tangent we could either,

find another tangent from the same external point and prove they have the same length (probably not for this question)

Prove that the tangent and a radius at the point of contact has an angle of 90 degrees (we dont know the center but can find it because we have 3 concurrent points and hence the center is at the intersection of the perpendicular bisectors, however proving that it exists at T could be a problem)

Or prove that we have angles between the chord and the tangent = in an opposite segment (similarly).

Or we could try to prove that the distance from the other circle's edge is greater then the radius at all but T



It might be easiest to draw up a tangent at T to one of the circles draw a concurrent triangle using the three points in both and then try to prove using trig and other circle props that it must also be a tangent to the other circle

But then again the information about the other two circles would be crucial somehow...

 

[Xayma]

I might have the answer to it but its going to be a bugger to try and explain through typing but anyway. (ie I have 4 variables)
(Im using < for angle)
Let the Circle containing the points P, Q and T be circle M
Let the Circle containing the points R, S and T be circle N

Ok Let < PQT=a
Let < TQS=b
Let < SRT=c
Let < PRT=d
< TSQ= a (angle between tangent (PQ) and chord (QS) equals angle in opposite segment)
< RST=b (similarly)
< TPQ=d (similarly)
< TPR=c (similarly)

Now PQRS is a quadiralateral therefore 2(a+b+c+d)=360 degrees
therefore a+b+c+d=180 degrees.

Construct a tangent at T to circle N.

Let this tangent be UV.

Now UTR=b ( < between tangent and chord=angle in opposite segment)

Therefore < PTU=a
( < sum of Triangle PTR, a+b+c+d=180 degrees)

Since < PQT=a
UV is also a tangent to the circle M (angle between tangent and chord=angle in opposite segment)

Therefore circle M and circle N touch at point T (common tangent at that point).

If you dont understand any step tell me and I will construct a diagram for it.

 

[Faera]

Hey

I don't really understand proofs unless they're algebraic when I'm reading online, so what I'll do is print yours of, Xayma, and take a look at it tonight so I have a hard copy with me, (and there's also the bonus of being able to look at circles, not ellipses) and then I'll get back to you sometime tomorrow- - thanks for all the help =)

P.S. GC, i'll take a look at yours, too. =)

Thanks, again!
(yayyyyy)