# the angle of A

#### shahar

##### Junior Member
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

##### Super Moderator
Staff member
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

##### Junior Member
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

##### Senior Member
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

##### Junior Member
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

##### Elite Member
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

##### Elite Member
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

##### Junior Member
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

##### Elite Member
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

##### Elite Member
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.