1. Given a line AB with points P and Q in the same half plane determined by line AB.

Desired path:

Hint:

Now if we reflect P and Q in AB, we get an isosceles triangle PP'Q'Q with AB as its axis. Construct the diagonals of the trapezoid and their intersection will be on AB at point X.

Since AB is the axis of reflection, it bisects the angles.

GSP file? Click

HERE.

Note that in this problem, we were NOT given that the path from P to X to Q was the shortest such path. In the link to the Minimal Path Problems, the task will be to find the shortest distance.

2. Given a line AB with points P and Q in the same half plane determined by line AB.

Hint?