How to Solve Complex Euclidean Geometry Proofs?

[Ameer Bux]

Homework Statement

write the proof

Homework Equations

none

The Attempt at a Solution

I've tried 5 times, got nowhere

 

[Buzz Bloom]

Hi Ameer Bux:

Problem 7 is tricky because the figure is not drawn in a manner consistent with the statements about it. You might find it helpful to redraw it so that SB is a diameter of the circle ABC. At least that's how I interpret the text:"SB bisects ABC."

Problem 8 is easier. I suggest you look up

https://en.wikipedia.org/wiki/Exterior_angle_theorem .​

Read about proposition 1.32.

Hope this helps.

Regards,
Buzz

 

[mfig]

What do the numbers (1&2) in the diagram indicate? Are all angles labeled 1 supposed to be equal?

 

[LCKurtz]

Hi Ameer Bux:

Problem 7 is tricky because the figure is not drawn in a manner consistent with the statements about it. You might find it helpful to redraw it so that SB is a diameter of the circle ABC. At least that's how I interpret the text:"SB bisects ABC."

I think it might mean that line SB bisects angle OBC. Here's a picture drawn to scale assuming that:

 

[LCKurtz]

@Ameer Bux I have verified that the figure should look like I drew in the post above with SB bisecting angle ABC. As a hint towards a proof I would remind you that angles on circles that subtend equal arcs are equal.

 

[LCKurtz]

What do the numbers (1&2) in the diagram indicate? Are all angles labeled 1 supposed to be equal?

Ameer appears to have disappeared, which is annoying enough. But to answer your question, looking at his second picture, I think when he as a 1 and 2 at vertex A that it is a very awkward notation where it would have been much better to call them A1 and A2. Similarly the 1 and 2 at vertex R refer to angles better notated as R1 and R2. So all those 1's and 2's are different. Awful notation in his pictures.

 

[LCKurtz]

Problem 8 is easier. I suggest you look up

https://en.wikipedia.org/wiki/Exterior_angle_theorem .​

Read about proposition 1.32.

Hope this helps.

Regards,
Buzz

Buzz, do you have an argument for problem 8? I don't see it and I have my doubts it is even true. And I don't see proposition 1.32 at that link. ??

 

[Buzz Bloom]

Buzz, do you have an argument for problem 8?

Hi LCKurtz:

Thank you for noting my error. Somehow I read the handwritten question as referring to R₁ rather than B₁. Sorry for my carelessness. Just another of my senior moments.

And I don't see proposition 1.32 at that link. ??

From the link:

... the term "exterior angle theorem" has been applied to a different result,[1] namely the portion of Proposition 1.32 which states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.​

Regards,
Buzz

​

 

[LCKurtz]

Hi Buzz. I wasn't trying to point out errors though, I'm really wondering if his Problem 8 is true. So I was hoping you had a proof. I can't figure out a proof and I drew what I consider to be a reasonably accurate figure which leads me to believe it may be false.

 

[haruspex]

Hi Buzz. I wasn't trying to point out errors though, I'm really wondering if his Problem 8 is true. So I was hoping you had a proof. I can't figure out a proof and I drew what I consider to be a reasonably accurate figure which leads me to believe it may be false.

Maybe I am missing something but it looks quite easy.
I sent details to Buzz in a private conversation, but for some reason it would not let me add you as a recipient. You must be in a higher astral plane. I thought it premature to post what might be a solution to the thread.

@Ameer Bux , what do you know about opposite angles of a quadrilateral whose vertices lie on a circle?

 

[atom jana]

Homework Statement

write the proof

Homework Equations

none

The Attempt at a Solution

I've tried 5 times, got nowhere

For the first problem, the theorem which states that equal arcs subtend equal angles on a circle is applicable.

 

[Ameer Bux]

Hi guys, thanks a lot for the help. I've solved both problems. I am going to post the solutions later to this thread. Much appreciated

 

[ciphone]

For the first problem, the theorem which states that equal arcs subtend equal angles on a circle is applicable.

I really have to learn geometry. I look foolish with a math degree without knowing geometry

 

[LCKurtz]

For the first problem, the theorem which states that equal arcs subtend equal angles on a circle is applicable.

You mean like what was mentioned in post #5?

 

[atom jana]

You mean like what was mentioned in post #5?

Yes, I see you had already mentioned it.