Question

Совпадают ли медиана и биссектриса треугольника? В каких случаях она совпадает или не совпадает? Как их отличить друг от друга???

Answer 1

Отрезок,  соединяющий вершину треугольника с серединой противоположной стороны,  называется медианой треугольника.  Отрезок,  который делит пополам угол,  из которого он выходит,  является биссектрисой угла треугольника.  В  общем случае эти отрезки не совпадают,  только в равнобедренном треугольнике,  если они проведены из вершины к его основанию,  то биссектриса и медиана  - один и тот же отрезок.  В  равностороннем треугольнике  эти отрезки совпадают  при проведении из всех трех вершин.

Answer 2

Mediana and Bisector of a Triangle

The median and bisector of a triangle are two different geometric concepts that are not the same.

Mediana: - The median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. - In other words, a median divides a side of a triangle into two equal segments, with one segment being twice as long as the other. - A triangle has three medians, each connecting a different vertex to the midpoint of the opposite side. - The three medians of a triangle always intersect at a point called the centroid. - The centroid is the center of mass of the triangle and is located two-thirds of the way along each median.

Bisector: - The bisector of a triangle is a line or line segment that divides one angle of the triangle into two equal angles. - In other words, a bisector divides an angle into two congruent angles. - A triangle has three angle bisectors, each dividing a different angle into two equal parts. - The three angle bisectors of a triangle always intersect at a point called the incenter. - The incenter is the center of the inscribed circle of the triangle, which is the largest circle that can fit inside the triangle.

Cases of Coincidence or Non-Coincidence

The medians and bisectors of a triangle do not generally coincide. However, there are certain cases where they can coincide:

1. Equilateral Triangle: In an equilateral triangle, all three sides are equal in length, and all three angles are equal. In this case, the medians and bisectors coincide. The medians are also the angle bisectors, and they all intersect at the centroid and incenter of the triangle.

2. Isosceles Triangle: In an isosceles triangle, two sides are equal in length, and two angles are equal. In this case, the medians and bisectors do not generally coincide. The medians still intersect at the centroid, but the bisectors do not coincide with the medians.

3. Scalene Triangle: In a scalene triangle, all three sides and angles are different. In this case, the medians and bisectors do not generally coincide. The medians still intersect at the centroid, but the bisectors do not coincide with the medians.

Distinguishing Between Medians and Bisectors

To distinguish between medians and bisectors in a triangle, you can consider the following:

- Medians divide the sides of a triangle into two equal segments, while bisectors divide the angles of a triangle into two equal angles. - Medians always intersect at the centroid, while bisectors always intersect at the incenter. - Medians are line segments, while bisectors are lines or line segments. - Medians are used to find the center of mass of a triangle, while bisectors are used to find the center of the inscribed circle of a triangle.

Remember that the medians and bisectors of a triangle are distinct concepts with different properties and intersections.