The Chicken Feet Theorem means that the distance from the intersection of the bisector of an interior angle of a triangle and its circumcircle to the other two vertices and the distance to the incenter and circumcenter are equal. This theorem is also known as the chicken feet theorem.

The chicken feet theorem has wide applications in triangles. For example, the chicken's foot theorem can be used to prove the properties of the incenter and paracenter of a triangle, or when solving problems related to triangles, the chicken's foot theorem can be used to find the incenter and paracenter to solve the problem. In addition, the chicken feet theorem can also be used to solve some problems related to the circumcircle of a triangle, such as determining the radius of the circumcircle of a triangle.

Proof process of the Chicken Feet Theorem:

1. First, it is known that the incenter of triangle ABC is l, the circumcenter of ZA is J, and the extension line of AI intersects the circumcircle of the triangle. Yu K.

2. Then, according to the angle bisector theorem, KI=KJ. According to the conclusion of the chicken feet theorem, KI=KJ=KB=KC.

3. Finally, according to the triangle midline theorem, KI=KB=KC, it is proved.

Steps to use the chicken feet theorem to solve the circumcircle problem of a triangle:

1. Find the bisector of an interior angle of the triangle, and determine the intersection point of the circumcircle of the angle, recorded as a point A.

2. Connect the two vertices of the angle to get two sides AB and A.

3. Using the chicken feet theorem, it can be concluded that the distance from point A to these two sides is equal, recorded as AD.

4. Extend these two sides to point E and point F respectively, so that AE=AF, connect EF, and obtain a straight line.

5. Pick any point G on the straight line EF and connect GB, GA, and GC.

6. Using the chicken feet theorem, it can be concluded that the distances from point G to these three vertices are equal, that is, the incenter and circumcenter of triangle ABC are on the straight line EF.

7. Since the incenter and circumcenter are on the same straight line, the circumcenter of triangle ABC is on the straight line EF. Through the above steps, we can find the circumcenter of triangle ABC.