Constructing a Homothety 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!

Consider the following figure:

Here are the steps of construction you have to follow:


*

*Construct the angle bisector of $k$ and $l$

*Take an arbitrary point (red thick) on the angle bisector.

*Drop a perpendicular to $l$ from the red point.

*Draw the red circle.

*Draw a line through $P$ and the intersection of $k$ and $l$. 

*This line will intersect the red circle.

*Connect this latter intersection point with the center of the red circle.

*Draw a parallel with this latter line through $P$.

*This parallel will intersect the angle bisector of $k$ and $l$.

*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...
