Circle Geometry Proof (Need someone to double check please)

[Tarmac27]

Hello,

I just want some input on this proof I did just to get some clarity. Anyway,

I started by extending PT to a point on the circle F, making PF a chord.

Then I extended another line from C that was tangent at F. So CP = CF since they are both tangents from same point.

Since, Triangle FTC and Triangle CTP share 90 degrees and 2 common sides they are congruent (RHS), so PT = TF

CP² = CB * CA (circle theorem)

and

CT² = CP² - TP²
= (CB * CA) - TP²

then,

TP * TF = TA * TB (circle theorem) (Recall PT = TF)
TP * TP = TA * TB
TP² = TA * TB

Hence,

CT² = (CB * CA) - (TA * TB)

Q.E.D

Im just not sure about the first step of extending PT to F and then extending the tangent. Is this something you can do?

Thanks

 

[Dr.Peterson]

I started by extending PT to a point on the circle F, making PF a chord.

That's valid; what do you need to prove to show that it is? What reason do you have to question it?

Then I extended another line from C that was tangent at F.

This is true; but do you need to prove tangency?

Triangle FTC and Triangle CTP share 90 degrees and 2 common sides they are congruent (RHS)

They are not named properly to call them congruent. Which vertices correspond?

Everything else looks good.

 

[Tarmac27]

Thanks for the reply,

1. To prove PF is a chord would I first have to show that TF is perpendicular to AB at T and since PF and TF are seperated by a diamter they are equal?

2. Taking what I said from "1." if you have already shown that TF and PF are equal then you would have congruency between the same two triangles again which would show that CF = CP hence proving tangency?

3. My mistake should it be triangle CFT is congruent to triangle CPT?

 

[Dr.Peterson]

1. To prove PF is a chord would I first have to show that TF is perpendicular to AB at T and since PF and TF are seperated by a diamter they are equal?

To show it's a chord, you only nee to show its endpoints are on the circle! And to obtain a chord by extending PT, you need to show that T is in the interior of the circle. You're showing additional facts here.

2. Taking what I said from "1." if you have already shown that TF and PF are equal then you would have congruency between the same two triangles again which would show that CF = CP hence proving tangency?

I would prove that CF is tangent by showing that it is perpendicular to OF. Do you have a theorem that if a segment is congruent to a tangent, it is itself tangent? You might.

3. My mistake should it be triangle CFT is congruent to triangle CPT?

Right.

 

[Tarmac27]

1. Ah ok, so you could prove that F is on the circle by showing that angle AFB is 90 degrees and it is given in the question that T is inside the circle.

2. Yes, I have the thoerem that says "a tangent drawn to a circle is perpendicular to the radius at the point of contact". Once you have proven CF is tangent hence CF =CP then you would still use the congruence of the triangles to get that PT = TF?

 

[Tarmac27]

Hang on, if P and F are the same distance away from A, then AF = AP. So triangles AFT and APT are congruent hence FT = PT, wouldn't this be better since it elimates the need of the CF tangent?

 

[Dr.Peterson]

I think you missed the point of my first comment. I was commenting on this:

I started by extending PT to a point on the circle F, making PF a chord.

Im just not sure about the first step of extending PT to F and then extending the tangent. Is this something you can do?

What I said was:

That's valid; what do you need to prove to show that it is? What reason do you have to question it?

You asked if it's valid to extend PT to make a chord. Of course you can -- as long as there is another point on the circle in that direction. I'm asking why in the world you think that might not be possible.

Or is what you're asking about, whether you can make a tangent at F? That's the next step, which I commented on separately.

You seem to think I was commenting on something else; but I'm not sure what that is!

Part of the problem is that you have not really been stating this carefully as a proof. Did you really mean only what you said, extending PT to meet the circle at F? Then you don't need to prove that F is on the circle, because it is by definition! Or did you mean that you extended it so that PT = TF? In that case, you do.

2. Yes, I have the thoerem that says "a tangent drawn to a circle is perpendicular to the radius at the point of contact".

But that's not what I said! I asked,

Do you have a theorem that if a segment is congruent to a tangent, it is itself tangent?

Do you not see the difference? We need to be very careful with wording in proofs.

Maybe you need to restate the entire proof, as if you were submitting it to your teacher, so we can be sure exactly what you are saying. But don't forget: I said that most of your work is good, and am only commenting on the details that are a little off.

 

[Tarmac27]

Sorry my mistake.

Yes, I see how those two different scenarios with the chords differ completely. In my proof I extended to F and then proved that PT = PF but I see how first stating PT = PF then proving F lies on the circle also works. I haven't heard of that theorem you stated, I just got confused because you said that you would prove CF by showing that angle OFC is 90 degrees.

I think that clears up all my confusion though, I appreciate you taking time to help improve upon these proofs.

Thanks