Constructing a Homothety

Question

This exercise has me stumped. I am meant to apply concepts concerning homotheties with circles to solve it. The problem states:

Given halflines k, l starting at a common point (let's call this point V), and a point P inside the angle formed by k and l, construct a circle through P tangent to k and l.

I tried multiple approaches to constructing this circle however I have yet to solve the problem.

Any help is much appreciated!

Answer 1

Consider the following figure:

Here are the steps of construction you have to follow:

1. Construct the angle bisector of $k$ and $l$
2. Take an arbitrary point (red thick) on the angle bisector.
3. Drop a perpendicular to $l$ from the red point.
4. Draw the red circle.
5. Draw a line through $P$ and the intersection of $k$ and $l$.
6. This line will intersect the red circle.
7. Connect this latter intersection point with the center of the red circle.
8. Draw a parallel with this latter line through $P$.
9. This parallel will intersect the angle bisector of $k$ and $l$.
10. This intersection point will be the center of the circle wanted.

Note that there is another circle through P that is tangent to $k$ and $l$. We could have chosen the other intersection point on the red circle...