Tangent problem of 1. circle: Tangent length theorem can help us solve problems related to circle, such as determining the length of tangent and judging whether two tangents are equal.

2. Geometric modeling: In geometric modeling, the tangent length theorem can be used to calculate and determine the tangent length of the object surface, thus helping us to accurately model and design.

3. Optical problems: In optical problems, the tangent length theorem can be used to calculate the propagation path and angle of light, thus helping us to understand the behavior and properties of light.

4. Engineering survey: In engineering survey, the tangent length theorem can be used to calculate and determine the distance and position relationship between the measuring point and the target, thus helping us to carry out accurate measurement and positioning.

5. Mathematical modeling: In mathematical modeling, the tangent length theorem can be used to establish mathematical models, thus helping us to solve various practical problems, such as the trajectory of object movement and the tangent equation of curve.

Summary:

Tangent length theorem is one of the commonly used theorems to solve geometric problems. It is used for tangent problem of circle, geometric modeling, optical problem, engineering measurement and mathematical modeling.

And has been widely used in many fields. Through the understanding and application of tangent length theorem, we can better solve practical problems and improve the accuracy and efficiency of solving problems.

First, the concept of tangent length theorem

The tangent length theorem means that the product of the tangent drawn from a point outside the circle and the radius is equal to the square of the distance from the tangent point to the center of the circle. In other words, if a tangent is drawn from a point outside the circle, the product of the tangent and the radius is equal to the square of the distance from the tangent point to the center of the circle.

Second, the proof of tangent length theorem

In order to prove the tangent length theorem, geometric reasoning and some basic geometric theorems can be used. First, we can form a right triangle by connecting the center of the circle, the tangent point and a point on the circle. Then, using Pythagorean theorem and similar triangles's properties, the conclusion of tangent length theorem is obtained.