Tangent and tangent length are two different concepts. The tangent is a straight line and cannot be measured. Tangent length is the length of a line segment. The two endpoints of this line segment are a point outside the circle and a tangent point, which can be measured.
The nature of tangent: tangent and circle have only one common point; The distance between the tangent and the center of the circle is equal to the radius of the circle; The tangent is perpendicular to the radius through the tangent point.
The discoverer of tangent theorem should be Miller. Inference of secant theorem: the product of two secant lines of a circle from a point outside the circle to the intersection of each secant line and the circle is equal. Secant theorem reveals the relationship between tangent and secant when they are drawn from a point outside the circle.
Emphasis: tangent length theorem and its application. Because the tangent length theorem once again embodies the symmetry of the circle, it provides a theoretical basis for proving the equality of line segments, angles, arcs and vertical relations. It belongs to tool knowledge and is often used, so it is the focus of this section.
Difficulties: Proof and calculation of tangent length theorem. For example, the third exercise on page 120 not only applies the tangent length theorem, but also applies the knowledge of understanding equations. It is a comprehensive problem of algebra and geometry, and students often cannot connect knowledge well.