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Math Help - Prove that there is a unique circle...

  1. #1
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    Prove that there is a unique circle...

    Let A be a point on a line l, and let B be a point that does not lie on l. Prove that there is a unique circle that contains A and B and is tangent to l.

    This is obviously true, but I am getting nowhere. Can I have any help please?
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  2. #2
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    Re: Prove that there is a unique circle...

    Quote Originally Posted by TimsBobby2 View Post
    Let A be a point on a line l, and let B be a point that does not lie on l. Prove that there is a unique circle that contains A and B and is tangent to l.

    There a unique ray \overrightarrow {AP} that is perpendicular to \ell that is in the B\text{-side}\cup\ell.

    Let \alpha be the perpendicular bisector of \overline {AB} .

    Can show that \alpha\cap\overrightarrow {AP}\ne\emptyset~?

    Is that point the center of the circle?
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  3. #3
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    Re: Prove that there is a unique circle...

    What do you mean the B side unioned with l? That's more confusing to me really, could you be a little more clear?
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    Re: Prove that there is a unique circle...

    Quote Originally Posted by TimsBobby2 View Post
    What do you mean the B side unioned with l? That's more confusing to me really, could you be a little more clear?
    I assumed you were in an axiomatic geometry course.
    But I guess you are not.

    There is a unique plane, \Pi, determined by \ell~\&~B

    Look at the unique perpendicular to \ell at A that is in \Pi.

    That perpendicular will intersect the perpendicular bisector of \overline {AB} .

    That is the center of your circle.
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