Proof:

To prove that the perpendicular at the point of contact to the tangent to a circle passes through the center, we can use the properties of tangents and the fact that the radius of a circle is perpendicular to the tangent at the point of contact.

Let's consider a circle with center O and radius r. Let P be a point on the circle, and let the tangent to the circle at P intersect the perpendicular at the point of contact at point Q.

Property 1: The tangent to a circle is perpendicular to the radius at the point of contact.
This property states that the tangent line at a given point on a circle is perpendicular to the radius of the circle at that point.

Property 2: If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line as well.
This property states that if we have two parallel lines and a third line that is perpendicular to one of the parallel lines, then it is also perpendicular to the other parallel line.

Proof:
1. Draw the circle with center O and radius r.
2. Draw the tangent line to the circle at point P.
3. Draw the radius OP.
4. By Property 1, we know that the radius OP is perpendicular to the tangent line at point P.
5. Draw a line perpendicular to the tangent at point P, and let it intersect the radius OP at point Q.
6. By Property 2, we know that the line through Q is also perpendicular to the radius OP.
7. Since the line through Q is perpendicular to the radius OP and passes through Q, it must pass through the center O as well.
8. Therefore, the perpendicular at the point of contact to the tangent to a circle passes through the center.

Hence, the perpendicular at the point of contact to the tangent to a circle always passes through the center of the circle.