Today's lesson we are going to find the tangent line from a cicle to abitrary point outside of the circle. This construction will be done using only a compass and a straight edge.
You can follow the construction with the sketches below. The markings in red are the construction each step.
Start by drawing a cirlce with your compass. Label the center A and the circle c1. Next, select an arbitrary point outside of the circle, label this P.
Using your straight edge, connect the center A with the point outside the circle, P. This gives us the line segment AP. Using your compass, conctruct a perperndicular bisector of AP (This is shown below by the dashed line). The intersection of the two lines is the midpoint of AP. Label this point M.
Measure the distance from M to P with your compass. Use this distance as the radius,with your compass, construct a circle with M as the center. Your cicle should go through points A and P as shouwn below. Label this circle c2. Label the intersection points of c1 and c2 point H and point G.
Finally, using your straight edge, draw a line through points P and H. Do the same with points P and G (extend the lines through the points). These lines of intersection are the tangents to our original circle C1.
You now have a circle and a line tangent to the cicle and a point outside of the cirlce.