Problem: Construct a tangent line to a given circle passing through a given point outside the circle. Euclid's solution to this problem is demonstrated using GSP.
Start with circle C2 with center E, and point A. We want a line through A and tangent to C2.
Construct line AE, then point D at the intersection with C2. Construct line j through D and perpendicular to AE.Construct circle C1, center at E and radius AE.
Construct point F at the intersection of j and C1.Construct line EF, then point B at the intersection of EF and C2.
Construct line AB.
Now triangle AEB is congruent to triangle FED
by SAS (they share angle E, FE = AE, and DE = BE).By definition,
angle FDE is 90 degrees, so angle ABE is 90 degrees and AB is
tangent to C2.