Assignment 7
-Problem-
Given a line and a circle with center K. Take an arbitrary point P on the circle. Construct two circles tangent to the given circle at P and tangent to the line.
-My Solution-
The basic idea of the solution is that the center of the circle that is tangent to the given circle at P should be on the line that pass through the points P and the center of the given circle and should be the same distance from the given point P and the given line.
First Step.
Draw a line L that passes through the center of the given circle (say 'C') and the point P and the tangent line of the circle C at the point P. The circle that is tangent to the circle C is also tangent to the tangent line of C at P. Since a line drawn perpendicular to a tangent at the point of contact with a circle passes through the center of the circle, the center of the circle tangent to C should be on the line L.
Next Step.
Now we should make the tangent circle contact with the given line.
Our goal is to find the point that is on the line passing through P and the center of C and that is equidistance from P and the given line.
To attain this goal, we need to make a isosceles triangle. Draw a circle whose center is the point Q that the intersection of the tangent line at P of C and the given line and whose radius is the distance between point P and Q, and let the intersection of the circle and the given line be R. Then, the triangle PQR is an isosceles triangle.
To find the point on the line passing through P and the center of C, draw a line passing through the point Q and the midpoint of P and R, and let the intersection of a new line and the line passing through P and the center of C be T. Then, the triangle TPR is also a isosceles triangle, and TP=TR.
Final Step.
Draw a circle whose center is the point T and whose radius is the distance between T and R.
Then, this circle is tangent to both the given circle C and the given line.
Now, explore the tangent circle as the point P moves along the given circle using file below.
If you want to see the GSP file, click HERE!