Demonstration for constructing a tangent to a circle through a given point
on it, with a straightedge only
Description of the construction
Let A be the point on the circle through which a tangent shall be drawn. Select any five points on the circle B, C, D, E and F. By using the GeoGebra-Dynamic Geometric Software, construct the chords AC, AD, BD, BE and CE.
Point K denote the point of intersection of the chords AC and BD.
Point L denote the point of intersection of the chords AD and EC.
Connect the points K and L by a straight line. The continuation of the straight line LK intersects the continuation of the chord BE at the point P. The straight line connecting the points P and A is the sought tangent AP.
As the point F is dragged toward the point A, the location of the points P, K, L remain on one straight line remain on the same one straight line. When point F coincides with the point A, the hexagon degenerates and the line FP becomes a tangent.
Note: if the straight lines KL and BE are parallel or almost parallel, its need to change slightly the position of point B and continue the construction as described.

Demonstration for constructing a tangent to a circle through a given point
on it, with a straightedge only
Description of the construction
Let A be the point on the circle through which a tangent shall be drawn. Select any five points on the circle B, C, D, E and F. By using the GeoGebra-Dynamic Geometric Software, construct the chords AC, AD, BD, BE and CE.
Point K denote the point of intersection of the chords AC and BD.
Point L denote the point of intersection of the chords AD and EC.
Connect the points K and L by a straight line. The continuation of the straight line LK intersects the continuation of the chord BE at the point P. The straight line connecting the points P and A is the sought tangent AP.
As the point F is dragged toward the point A, the location of the points P, K, L remain on one straight line remain on the same one straight line. When point F coincides with the point A, the hexagon degenerates and the line FP becomes a tangent.
Note: if the straight lines KL and BE are parallel or almost parallel, its need to change slightly the position of point B and continue the construction as described.