Lengths of line secgment that bisect angle

applying the above fact to each of these smaller triangles, we have:t_1 + t_2 > a_3, t_1 + t_3 > a_2, and t_2 + t_3 > a_1adding these three inequalities together, we get:2(t_1 + t_2 + t_3) > (a_1 + a_2 + a_3)dividing both sides by 2, we have:(t_1 + t_2 + t_3) > (a_1 + a_2 + a_3)/2since this is true for any triangle, we can apply it to our specific triangle and get:1/t_
  • #1
ER901
2
0

Homework Statement


Let t_i be the lengths of the line segments that bisect angle Ai of a triangle. The segments go from A_i to the opposite side. Let a_i be the lengths of the sides opposite angle A_i. Prove the following inequalities:

[tex]\sum[/tex]from i=1to 3 1/t[tex]_{}i[/tex] > [tex]\sum[/tex] from i=1 to 3 1/a[tex]_{}i[/tex]

[tex]\sum[/tex]from i=1 to 3 t[tex]_{}i[/tex] < [tex]\sum[/tex] from i=1 to 3 a[tex]_{}i[/tex]


The Attempt at a Solution


I'm not sure where to start.. but this is what I have in mind:
a_2, a_3 > t_1 ; a_1, a_3 > t_2 and so on...

but then I don't know what to do next
 
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  • #2
this is what I would try:
use the fact that the sum of length of 2 sides of any triangle is bigger than the remaining side
also, note that the three bisecting line segments cut the original triangle into 6 smaller triangles
 
  • #3
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As a scientist, my response would be to suggest approaching this problem using the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, we can use this theorem to show that the sum of the lengths of the bisecting line segments (t_i) must be greater than the sum of the lengths of the sides opposite those angles (a_i).

To prove the first inequality, we can start by considering the triangle formed by the three bisecting line segments. By the triangle inequality theorem, we know that the sum of any two sides of this triangle must be greater than the length of the third side. Therefore, we can say that:

t_1 + t_2 > t_3
t_1 + t_3 > t_2
t_2 + t_3 > t_1

Adding these three inequalities together, we get:

2(t_1 + t_2 + t_3) > t_1 + t_2 + t_3

This simplifies to:

t_1 + t_2 + t_3 > 0

Since all the lengths in this equation are positive, we can divide both sides by t_1t_2t_3 to get:

1/t_1 + 1/t_2 + 1/t_3 > 0

This is equivalent to the first inequality we need to prove. Therefore, we have shown that the sum of the reciprocals of the bisecting line segment lengths is greater than the sum of the reciprocals of the side lengths opposite those angles.

To prove the second inequality, we can use a similar approach. This time, we consider the triangle formed by the three sides opposite the angles (a_i). Again, by the triangle inequality theorem, we can say that:

a_1 + a_2 > a_3
a_1 + a_3 > a_2
a_2 + a_3 > a_1

Adding these three inequalities together, we get:

2(a_1 + a_2 + a_3) > a_1 + a_2 + a_3

This simplifies to:

a_1 + a_2 + a_3 > 0

Again, since all the lengths in this equation are positive, we can divide both sides by a_
 

1. What is a line segment that bisects an angle?

A line segment that bisects an angle is a line that divides the angle into two equal parts, creating two congruent angles. This line segment passes through the vertex of the angle.

2. How do you find the length of a line segment that bisects an angle?

To find the length of a line segment that bisects an angle, you can use the angle bisector theorem, which states that the length of the line segment is equal to the ratio of the lengths of the two sides of the angle, multiplied by the length of the side opposite the angle.

3. Can a line segment bisect more than one angle?

Yes, a line segment can bisect more than one angle. If the line segment passes through the vertex of multiple angles, it will bisect all of those angles.

4. What is the relationship between the bisector of an angle and the perpendicular bisector of a line segment?

The bisector of an angle and the perpendicular bisector of a line segment are both lines that divide a geometric figure into two equal parts. However, the bisector of an angle divides an angle into two equal angles, while the perpendicular bisector of a line segment divides the line segment into two equal parts.

5. How can the length of a line segment that bisects an angle be used in solving geometric problems?

The length of a line segment that bisects an angle is often used in finding the area of a triangle or other geometric figures. It can also be used in solving problems involving angle measures and congruent angles.

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