the angle of A

shahar

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O - the centre of a circle
Rs - two radii
The original question: What is bigger AB or AC?
The answer: Depend on the inequality of the triangle theorem.
My question:
What is the domain of A?
A's size is from what size to what size...
 

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Subhotosh Khan

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O - the centre of a circle
Rs - two radii
The original question: What is bigger AB or AC?
The answer: Depend on the inequality of the triangle theorem.
My question:
What is the domain of A?
A's size is from what size to what size...

A is a point - it does not have size!

As it stands can be located anywhere on the plane of the circle and outside the circle.
 

shahar

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In Israel when we saying Angle A, we mean that the interior angle of A or CAB
because CB and AB is the size of that angle.
So, (1) Angle CAB not have a size at all.
(2)
or: it's has the range of size from the minimum size to the maximum size
What is the maximum size?
What is the minimum size?
 

Harry_the_cat

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O - the centre of a circle
Rs - two radii
The original question: What is bigger AB or AC?
The answer: Depend on the inequality of the triangle theorem.
My question:
What is the domain of A?
A's size is from what size to what size...
Assuming OBA is a straight line:

OC + CA > OA (the shortest distance between two points is a straight line).

Since OA =OB + BA, it follows that

OC + CA > OB + BA,

but OC =OB =R,

therefore CA > BA or AC > AB

(Note here I have used AC to represent the length AC, not the vector)

To answer your next question, the possible size of angle A depends on where A is in relation to the circle.

Clearly A>0 and the closer A is to the circle, the max value approaches 90 degrees. So I'd say 0<A<90 (degrees)
 
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shahar

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So. 0 is cleary to me

But Can it will be proven that The max. size of Angle A is 90 degree?
By Synthetic Geometry.
or by other way but please not vectors method because in vector I know the Idea I think.
 

Dr.Peterson

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O - the centre of a circle
Rs - two radii
The original question: What is bigger AB or AC?
The answer: Depend on the inequality of the triangle theorem.
My question:
What is the domain of A?
A's size is from what size to what size...
If it is known that A is outside the circle, then angle OAC < angle OBC; the latter is the base angle of an isosceles triangle, and therefore must be less than 90 degrees.
 

pka

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But Can it will be proven that The max. size of Angle A is 90 degree?
By Synthetic Geometry.
As was pointed out in the immediate previous post,#6, the triangle \(\displaystyle \Delta OBC\) is isosceles meaning that angle \(\displaystyle \angle OBC\) is acute.
Moreover, \(\displaystyle \angle OBC\) is exterior to \(\displaystyle \Delta ABC\) so by the exterior angle theorem \(\displaystyle m(\angle OBC)=m(\angle BCA)+m(\angle BAC)\)
That means that \(\displaystyle m(\angle B{\large\bf{A}}C)<\dfrac{\pi}{2}\)
 

shahar

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As was pointed out in the immediate previous post,#6, the triangle \(\displaystyle \Delta OBC\) is isosceles meaning that angle \(\displaystyle \angle OBC\) is acute.
Moreover, \(\displaystyle \angle OBC\) is exterior to \(\displaystyle \Delta ABC\) so by the exterior angle theorem \(\displaystyle m(\angle OBC)=m(\angle BCA)+m(\angle BAC)\)
That means that \(\displaystyle m(\angle B{\large\bf{A}}C)<\dfrac{\pi}{2}\)
What the meaning of the notion m(angle X)
What is m?
...m(Some Angle) I mean?
 

Dr.Peterson

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What the meaning of the notion m(angle X)
What is m?
...m(Some Angle) I mean?
This means "the measure of angle X" (that is, the number of degrees or radians in the angle). See here, and here. Not everyone uses this notation; it may be largely an American usage, but I am not sure.
 

pka

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What the meaning of the notion m(angle X)
What is m? ...m(Some Angle) I mean?
Because you used the term "synthetic geometry" I naturally assumed that you were literate in the subject.
In axiomatic geometry textbooks it is fairly standard to usee \(\displaystyle m(\angle A)\) for the measure of the angle.
 
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