Circle Geo (1 Viewer)

[Faera]

Hey guys-
These are out of the 4u Fitzpatrick book, ex 37(a)
If you haven't had a go at them you probably should try taking a look at them some time soon because they take aaagggesss to see.. (or it could've just been me).
Well, anyway, there are three that I can't get out, any help would be good
These questions are driving me absolutely insane!!
-.-'


1. PQ and RS are common tangents to 2 intersecting circles. If T is one of the points of intersection of the 2 circles, prove that the circles through T, P, and Q and through T, R, S touch each other.

2. A right angled triangle ABC with right angle A circumscribes a circle of radius r. Prove that r = ()(c + b a) where a, b, c are the length measures of the sides of the triangle. (that is, a is the side opposite angle A, b is the side opposite angle B, and so on)

3. Draw three circles such that each intersects the other two. Prove that the three common chords are concurrent.


*evil voice*
Have fun...

 

[CM_Tutor]

Question 3 was proven in the 1995 4u HSC, q 6(b), if someone wants a more structured question.

Faera, are you looking for some help, or providing a challenge, or both?

 

[Faera]

both.

 

[Grey Council]

both, me thinks.

edit:
were typing at same time

EDIT EDIT:
Question 1 is question 8 of fitzpatrick 37a
Question 2 is question 30 of fitzpatrick 37a
Question 3 is question 25 of fitzpatrick 37a

 

[Faera]

Mine popped up first, though... =D

Oh, and I'll hunt for my past papers book for that, thanks, CM.

 

[Grey Council]

cause you told me after you posted.

ANYWAY, the questions.

are these in the syllabus?

 

[Faera]

Well, we're s'posed to use c/g skill s from 3u to do them, so Im guessing yes.
But as to if well get asked them in tests Q3 was asked.. =D
But I think the 1st one is a bit too complicated. Not sure about the 2nd.

 

[ND]

Umm are you sure 2 is correct? Cos 2r=a, and if the identity is true then b+c=2a, which obviously isn't right. Unless i don't know what circumscribes means... (which is probably the case)

 

[Grey Council]

fair enough.

alrhough question 3 wasn't quite asked, was more, they led us through the question.

welcome back to the forums, btw.

 

[Faera]

Originally posted by ND
Umm are you sure 2 is correct? Cos 2r=a, and if the identity is true then b+c=2a, which obviously isn't right. Unless i don't know what circumscribes means... (which is probably the case)

Why is it obviously not right?
circumscribes = Makes a circle going through points A, B, C

 

[ND]

Originally posted by Faera
Why is it obviously not right?
circumscribes = Makes a circle going through points A, B, C

That means that a is the diametre, 2r. It's not correct cos b^2+c^2=a^2.

 

[Xayma]

Actually the way the question is phrased it implies that a circle is inside the triangle touching at only 3 points.

If you were to circumscribe the triangle the radius would be 1/2a (angle in semicircle) not 1/2(b+c-a)

I have nearly proved it I just need to finish off proving that I have a square in part of the diagram, (I might be able to use angle between tangent and chord but I dont know if it is an alternate segment since it only goes to the centre).

Actually I can use the radius vs tangent angle (at point of contact)=90 degrees.

Get Ready for long confusing proof (hmm I probably should do a diagram in paint to make it less confusing)

 

[Grey Council]

hrm, ND? not sure what your saying.

Originally posted by ND
That means that a is the diametre, 2r. It's not correct cos b^2+c^2=a^2.

you are right, a is the diameter, of length 2r. b^2 + c^2 = a^2. whats wrong with that?

 

[Faera]

Originally posted by ND
That means that a is the diametre, 2r. It's not correct cos b^2+c^2=a^2.

Sorry, I dont understand why it can't be correct.
I'm not really following. -.-'

Originally posted by Xayma
Actually the way the question is phrased it implies that a circle is inside the triangle touching at only 3 points.

If you were to circumscribe the triangle the radius would be 1/2a (angle in semicircle) not 1/2(b+c-a)

Perhaps we have to prove that a = (a + b)/2, then?

 

[ND]

Ok assume the identity to be correct:

r=(b+c-a)/2
2r=b+c-a
2a=b+c (cos a=2r)

but b^2+c^2=a^2 (pythag's theorem)
a=sqrt(b^2+c^2)
subbing into the identity:
sqrt(b^2+c^2)=b+c/2
which is only true for b=c=0.
.'. by contradiction, r != (b+c-a)/2

 

[Grey Council]

nice, well done ND.

so i think it is circumscribed IN the triangle. Xayma is right.

EDIT:
oo, well done, Xayma.

 

[Xayma]

Ok
Let the distance from the intercept of the circle and triangle on line "a" to the vertext at < C=d

Hence the distance from the intercept of the circle and triangle on line "b" to < C=d (Tangents from a common external point=in length at the point of contact)

Let the distance from the intercept of the circle and triangle on line "b" to the vertex at < A=f

Hence the distance from the intercept of the circle and triangle on line "c" to < A=f (Tangents from a common external point=in length at the point of contact) [Lot simpler on paper with all those as above signs]

Let the distance from the intercept of the circle and triangle on line "c" to the vertext at < B=e

Hence the distance from the intercept of the circle and triangle on line "a" to < B=e (Tangents from a common external point=in length at the point of contact)

Therefore

e+f=c --1
d+e=a --2
d+f=b --3

Adding 1 and 3

b+c=2f+e+d --4

Subtracting 2 from 4

c+b-a=2f

Hence 1/2(c+b-a)=f

Now the radius from the center of the circle forms an angle of 90 degrees with line f (both of them, since they stem from a common point) (Angle between a tangent and radius at the point of contact=90 degrees).

Therefore the radius=f (as the quadrilateral formed from the radius to the points of intercepts with the two lines of length f is a square (90 degree angles, adjacent sides=)

Therefore r=1/2(b+c-a)

 

[Xayma]

Oh yeah, heres the picture

 

[Faera]

Hmm *kicks self* -.- ive been doing the question WRONG!
Blah.
Sorry, ND, about that.
My bad.
Thanks for the proof, Xayma.
*gives cake to all*
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.

 

[Grey Council]

well done Xayma, and well pointed out ND.
I haven't done anything, so i don't deserve a cake.

But now, next question. actually, the frist one.
the last one is in the HSC exam, hrm?
so first one

for convenience, i'll post it here:
1. PQ and RS are common tangents to 2 intersecting circles. If T is one of the points of intersection of the 2 circles, prove that the circles through T, P, and Q and through T, R, S touch each other.

ND and CM_TUtor, if you figure out how to do it, can you give us hints, instead of the solution?