Circle Geometry Question (1 Viewer)

[Drongoski]

golden answer

+reppped

jetblack2007 is so outstandingly good in maths that I didn't bother to check his proof. I was more concerned about being behind him in my post (haha). But now on looking at his post again, I know his proof is wrong; i.e. AB does not bisect angle EAC I think his 2nd last line is not valid conclusion.

I then went to construct a large and fairly accurate diagram to see if AB bisects .. and lo & behold, as suspected it does not. I made angle BAC about 30 deg.

Then I realised I was being stoopid again. Since angle CAE is 90 deg, if AB bisects CAE, it means angle CAE must be 45 deg.; but angle BAE can be 22deg, 37 deg whatever; not necessarily 45 deg. That means I shouldn't need to bother with the large diagram.

 

[lyounamu]

jetblack2007 is so outstandingly good in maths that I didn't bother to check his proof. I was more concerned about being behind him in my post. But now on looking at his post again, I know his proof is wrong; i.e. AB does not bisect angle . . I think his 2nd last line is not valid conclusion.

I then went to construct a large and fairly accurate diagram to see if AB bisects .. and lo & behold, as suspected it does not. I made angle BAC about 30 deg.

Then I realised I was being stoopid again. Since angle CAE is 90 deg, if AB bisects CAE, it means angle CAE must be 45 deg.; but angle BAE can be 22deg, 37 deg whatever; not necessarily 45 deg. That means I shouldn't need to bother with the large diagram.

My thought exactly. But still cannot be bothered to check whether he is right or wrong @.@

 

[jet]

Lol. I am wrong. Sorry bout that.
I was really tired but oh well. The op has there answer. My proof was certainly not strong enough to hold up to rigorous argument.

So... this whole thing was all over nothing... lol.

 

[Drongoski]

Lol. I am wrong. Sorry bout that.
I was really tired but oh well. The op has there answer.

So... this whole thing was all over nothing... lol.

No need to apologise, It shows we all make mistakes. you have been putting up so many outstanding solutions.

 

[nrlwinner]

Sorry guys, there are too many different things going on and I still don't understand the solution.

 

[Timothy.Siu]

Sorry guys, there are too many different things going on and I still don't understand the solution.

what part dont u get

 

[GUSSSSSSSSSSSSS]

Sorry guys, there are too many different things going on and I still don't understand the solution.

just read drongoski's first post and that gives it

drongoski actually wrote up 2 solutions in that first post if i remember correctly

 

[nrlwinner]

Here's soln. But I don't know how to get diagram up.

Draw a circle centre o. Pick two points a & b on circle. Draw diameter thru A & O to meet circle at C. Draw tangent line at B and drop a perpendicular from A to this tangent at E. Draw lines AB and BC. Let F be any point on tangent opposite side of B from E .

Now angle OBE is a right angle, Therefore AE//OB.
Now angle CBF = angle CAB = @ say ( angle in alt segment)
Since triangle AOB is isosceles, angle OBA = angle BAO = @
Angle EAB = angle OBA = @ (alt angles , AE//OB)

Therefore angle EAB = angle BAC

Hence AB bisects angle CAE


QED

jetblack2007: very nice diagram; wish I knew how to do it.

Edit

Oh what an embarassingly long-winded solution. Here's revised version; forget F)

angle BAO = angle ABO (ABO isosceles)
But angle EAB = angle ABO (alternate angles, AE//OB)
Therefore angle OAB = angle EAB
Hence AB bisects angle CAE

How can you prove AE is parallel to OB

 

[hungwell1337]

How can you prove AE is parallel to OB

re-read the question

jetblack's diagram is wrong

 

[Drongoski]

1. tangent @ B is perp to the radius (i.e. OB) to B

2. AE is perp. to this same tangent

3. so aren't they parallel ?

 

[GUSSSSSSSSSSSSS]

How can you prove AE is parallel to OB

the tangent is right angle to the radius at B
and we are TOLD that AE is perpendicular to the tangent

sooooooooo corresponding angles are equal
which means they are parallel

 

[jet]

Ok, updated to go with Drongoski's solution:

 

[Timothy.Siu]

How can you prove AE is parallel to OB

because cointerior angles are 180'

 

[nrlwinner]

Ok, updated to go with Drongoski's solution:

Thank you very much. I get it now.
On a similar note, how many things do you need to prove to make 2 lines parallel (for example, in this diagram, is that 1 pair of co-interior angles adequate enough to say they are parallel?)p

 

[Drongoski]

i pair will do.

Oh! That's a fantastic diagram jetblack. Thanks for illustrating my soln.

 

[GUSSSSSSSSSSSSS]

Thank you very much. I get it now.
On a similar note, how many things do you need to prove to make 2 lines parallel (for example, in this diagram, is that 1 pair of co-interior angles adequate enough to say they are parallel?)p

yea, if u can prove ANY property of parallel lines, then they ARE parallel lines