This is the original phrasing of the question:
"Describe a circle to touch a given circle, and also to touch a given
straight line at a given point."
- A School Geometry H.S. Hall and F.H. Stevens p.184 q.9
See Figure 1. drawn in Geogebra
In the figure, suppose AB is the given line and the circle with center O and radius OP is the given circle.
This is part of a euclidean geometry textbook, so it would be preferred if the answer can be kept within the scope of euclidean geometry.
There are a few Theorems which I believe would help:
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If two circles touch one another, the centre, and the point of contact are in one straight line (Hence the centre must lie on a straight line from the given circle's centre). The difficulty is in ascertaining which radius is to be produced to find the centre of the required circle.
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For a circle to touch a straight line at a given point, the straight line must be perpendicular to a radius at that point. Hence the centre must lie on the perpendicular to the given straight line at the given point.
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Of course, radii of the same circle are equal. Hence PS = SR
I am quite stuck on this problem. Thank you very much for any help. They are greatly appreciated.
Best Answer
Let the point on the line be $P$ and the center and the radius of the circle be $O$ and $r$, respectively.
Begin by drawing a perpendicular line to the given line at point $P$.
On this perpendicular line, locate and mark point $F$ at a distance of $r$ from point $P$.
Next, construct the perpendicular bisector of line segment $OF$.
The intersection point of the perpendicular bisector with line $PF$ will serve as the center for the circle in question.
It's worth noting that there are two possible positions for point $F$, resulting in the possibility of two different circles.