Given a line, L and two points, A & B not on the line, construct a circle which contains both points and is tangent to

the given line.

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- Apr 6th 2013, 09:20 PMJammixConstruction help!
Given a line, L and two points, A & B not on the line, construct a circle which contains both points and is tangent to

the given line. - Apr 8th 2013, 10:11 PMibduttRe: Construction help!
- Apr 9th 2013, 12:41 PMMINOANMANRe: Construction help!
The lines AB and L intersect at a fixed point P.

The circle we search has centre at O . and the line L is tangent to this circle at the point Q.

According to a well known theorem of the Euclidean Geometry (PA) X (PB) = (PQ)^2 therefore PQ =SQRROOT[(PA) X (PB)]

this expression defines the position of the point of tangency Q along the line L . The centre of the circle is the intersection point O of the perpendicular bisector of AB and the perpendicular line to the line L at Q. OA =OB =OQ = r is the radius of the circle .

MINOAS - Apr 9th 2013, 01:22 PMPlatoRe: Construction help!

The original question contains a mistake.

What if your point is between $\displaystyle A~\&~B$, that is $\displaystyle A-P-B~?$ If so the no such circle exists.

So it must be stated that $\displaystyle A~\&~B$ are on the same side of $\displaystyle \ell$.

But then it may be that $\displaystyle \overleftrightarrow {AB} \cap \ell = \emptyset$, parallel. In which case $\displaystyle P$ does not exist.