Question

Сформулируйте и докажите свойство биссектрисы треугольника.

Answer 1

Пусть  BL  биссектриса угла  B  ,         B ∈ [AC]
S(ABL)/S(CBL) = (1/2AL*h)/(1/2CL *h)   = AL/CL        (1)
  с другой  стороны отношение  площадей этих треугольников равно
S(ABL)/S(CBL) = (1/2AB*BL*sinB/2)/(1/2CB*BL*sinB/2) = AB/CB     (2)
Из  уравнений   (1)   и  (2)  следует
AL/CL = AB/CB     [ отношение отрезков = отношению сторон  ]
  
Можно доказать  чисто геометрически

Answer 2

Property of a Triangle's Bisector

A bisector of a triangle is a line that divides one of the angles of the triangle into two equal angles and also divides the opposite side in the same ratio. In triangle ABC, let AD be the bisector of angle A, where D is on BC.

The property of a bisector of a triangle can be formulated and proved as follows:

1. Angle Bisector Theorem: The angle bisector theorem states that the bisector of an angle of a triangle divides the opposite side in the same ratio as the other two sides of the triangle. In other words, if AD is the bisector of angle A in triangle ABC, then the following proportion holds: - BD/DC = AB/AC

This theorem can be proved using the Law of Sines or by drawing a parallel line to one of the triangle's sides and using similar triangles. The proof involves establishing the equality of two ratios involving the sides of the triangle and the sines of the angles.

Source: 2. Converse of the Angle Bisector Theorem: If a point is in a triangle and divides one side of the triangle into segments that are proportional to the other two sides, then it lies on the bisector of the angle opposite the side.

This converse theorem essentially states that if a point D on BC satisfies the proportion BD/DC = AB/AC, then AD is the bisector of angle A.

Source: 3. Angle Bisector and Angles of the Triangle: The angle bisector divides the opposite side into segments that are proportional to the adjacent sides. Additionally, the angle bisector divides the opposite angle into two angles that are proportional to the adjacent angles.

This property can be used to solve various problems related to triangles and their angles.

Source: In summary, the bisector of an angle in a triangle has the property of dividing the opposite side in the same ratio as the other two sides of the triangle, and it also divides the opposite angle into two angles that are proportional to the adjacent angles. These properties are fundamental in the study and application of triangle geometry.

If you have further questions or need additional information, feel free to ask!