Sine theorem is a/sinA=b/sinB=c/sinC.

Because sin∠ABD=sin∠CBD.

Sin∠ADB=sin∠CDB (sine values of complementary angles are equal)

So AB/BC=AD/DC

Its nature is as follows:

The bisector of 1. can get two equal angles.

2. The distance from the point on the bisector of the angle is equal to both sides of the angle.

3. The three bisectors of a triangle intersect at a point, which is called the triangle center. The distance from the center of the triangle to three sides of the triangle is equal.

The two line segments formed by the opposite sides of the bisector of this angle are proportional to the two adjacent sides of this angle.

Extended data:

Angular bisector definition:

1. Draw a ray from the vertex of an angle (the line is inside the angle) and divide the angle into two identical angles. This ray is called the bisector of an angle.

2. The bisector of an angle is the locus of points in the shape of the angle that are equidistant from both sides of the angle from the shape.

The bisector of an angle of a triangle intersects the opposite side of the angle, and the line segment connecting the vertex of the angle and the opposite side is called the bisector of the triangle (also called the bisector of the inner angle of the triangle). By definition, the bisector of a triangle is a line segment. Because a triangle has three internal angles, it has three bisectors. The intersection of the bisectors of a triangle must be inside the triangle.

References:

The nature of angular bisector-Baidu Encyclopedia