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?
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?
There a unique ray $\displaystyle \overrightarrow {AP} $ that is perpendicular to $\displaystyle \ell$ that is in the $\displaystyle B\text{-side}\cup\ell.$
Let $\displaystyle \alpha$ be the perpendicular bisector of $\displaystyle \overline {AB} $.
Can show that $\displaystyle \alpha\cap\overrightarrow {AP}\ne\emptyset~?$
Is that point the center of the circle?
I assumed you were in an axiomatic geometry course.
But I guess you are not.
There is a unique plane, $\displaystyle \Pi$, determined by $\displaystyle \ell~\&~B$
Look at the unique perpendicular to $\displaystyle \ell$ at $\displaystyle A$ that is in $\displaystyle \Pi$.
That perpendicular will intersect the perpendicular bisector of $\displaystyle \overline {AB} $.
That is the center of your circle.