Day 2: Circle Tangents

Use GSP 4.0 to perform the following investigations.

Investigation 1:

Step 1: Construct two points. Label these points A and B. Construct the circle with center A that passes through B.

Step 2: Construct a point on circle A. Label this point C. Construct a point D somewhere on the exterior of circle A. Construct line CD.

Step 3: In all likelihood, line CD is a secant of circle A. Measure angle ACD. Drag point C until line CD becomes tangent to circle A. Observations?

Step 4: Prove your conjecture. Hint?

Extension: What would the converse of this theorem be? Is this converse true?

Investigation 2:

Using the results of investigation 1, draw a circle and contruct two lines that are tangent to this circle. Adjust your sketch until the tangents intersect at a point X. If the points of tangency are M and N, what appears to be true about MX and NX? Measure these values and try tragging different points.

Conjecture? Proof? Hint?