The problem is to find two different circles tangent to the following circle at point P and to the following line.
I need to find the center of the circle. Here is my construction:
First of all, I know that the center should be on the line which passes through the center of the given circle and the point P.
I should find such a point on this line that is equidistant from the point P and the given line.
I construct a line perpendicular to the given line that passes through the center of the given circle. I mark the intersection of this line with the given circle as point B.
Then I construct the line through B and P. I mark the intersection of this line with the given line as point C.
Finally, I construct the perpendicular line to the given line at point C and I propose that point D in the following drawing is the center of one of the tangent circles.
Why is D the center? To prove this I need to show that DC = DP
The following angles are equal: EBP = BPO = DPC
I need to show that angle DCP is also equal to these angles.
Look at angle PCE : It is complementary to both EBP (because BEC is a right angle) and to DCP (because DCE is a right angle).
So, DCP = EBP and hence DCP = DPC.
Therefore, DC = DP and D is the center of our tangent circle. Here it is:
Where is the second circle?
The center again should be on the line connecting the center of the given circle and the point P.
This time, I draw the line connecting P and A. A is one of the intersection points of the given circle and the line through O which is perpendicular to the given line.
I mark the intersection point of this line with the given line as J. Then, I draw a perpendicular to the given line at point J.
I propose that point K is the center of the second tangent circle.
To prove this, I need to show that KP = KJ.
I know that the following angles are equal:
OPA = OAP (Why?)
OAP = KJP (Remember that OA is parallel to KJ.)
So, OPA = KJP.
Hence, KP = KJ.
Here is the tangent circle:
You can click here to observe the tangent circles as point P is taken at different points on the given circle.
Is there any point P for which we can't find two different circles? (Yes.)