Proving ABC is Isosceles: Triangle ABC and Bisectors

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In summary, The conversation is discussing a problem where they need to prove that triangle ABC is isosceles. One person suggests using a picture and the Sine Rule to prove it, while another person talks about their experience with the problem and their difficulty in solving it. They also discuss the distinction between homework forums and the current forum, and clarify that the problem was not from a homework assignment. One person makes a mistake in their construction and realizes that they assumed AB as the hypotenuse, which is not correct. Overall, the conversation revolves around different approaches and thoughts on solving the problem.
  • #1
jdavel
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am i missing something, or is this really hard to prove? anyone know a proof? an elegant one?

Given:
triangle ABC
bisector of A intersects BC at D
bisector of B intersects AC at E
AE = BD

Prove: ABC is isosceles
 
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  • #2
Draw a picture, and apply the Sine Rule like possessed. Notice that the two angles at D have the same sine; this is also true for the two angles at E.
 
  • #3
Have the moderators given up on maintaining the distinction between these forums and the homework forums?
 
  • #4
I first encountered this problem almost eight years ago and on occasion still try to prove it. It serves as a kind of meditation, almost Zen in a way.

I did manage to prove it using coordinate geometry, but of course that's cheating. A paper and compass solution eludes me still. I'm of two minds as to whether I'd like to get a solution from this thread, or still try and solve it myself.
 
  • #5
DoDo,

That works. Nice!

P.S. Ben Niehoff, this really wasn't from a homework assignment. I ran across this problem years ago, could never solve it, and just happen to think of it a few days ago.
 
  • #6
jdavel said:
P.S. Ben Niehoff, this really wasn't from a homework assignment. I ran across this problem years ago, could never solve it, and just happen to think of it a few days ago.

Sorry, my mistake. But there have certainly been a lot of them in this forum lately. It gets annoying after a while.
 
  • #7
If we let [tex]\alpha[/tex] = ABE and [tex]\beta[/tex] = DAB, does then sin[tex]\alpha[/tex]=AE/AB and sin[tex]\beta[/tex]=BD/AB?
And since AE = BD, then [tex]\alpha[/tex] and [tex]\beta[/tex] are equal?

Am I making assumptions from my sketch here?
 
  • #8
Why sin alfa = AE/AB ? What the Sine Rule says is that (sin alfa) / AE = (sin angle AEB) / AB.
 
  • #9
I understand what you're saying, however, all I am using here is sin=opp/hyp. In my construction, sin(beta) = BD/hyp, and sin(alpha)=AE/hyp.

If they share the same hypotenuse (side AB in my post), then because AE=BE the two angles will be the same.

Here's my problem: I do believe I assumed AB is the hypotenuse for the two triangles being compared.
 
  • #10
In order to have an hypotenuse, you need a right angle somewhere.
 
  • #11
Right. I realize now the very obvious mistake I made. I chose to draw the triangle equilateral, giving me the right angles I needed. Of course, that breaks the rules from the get-go.
 

FAQ: Proving ABC is Isosceles: Triangle ABC and Bisectors

How do you prove that a triangle is isosceles?

To prove that a triangle ABC is isosceles, we need to show that two sides of the triangle are equal in length. This can be done by using the properties of bisectors, which divide a line segment into two equal parts. By showing that the bisectors of two sides of the triangle are equal, we can conclude that the two sides are also equal, making the triangle isosceles.

What are the properties of bisectors?

Bisectors are lines or line segments that divide a line segment into two equal parts. They also create two equal angles at the point where they intersect the line segment. In a triangle, the bisector of an angle also divides the opposite side into two equal parts.

What is the relationship between bisectors and isosceles triangles?

Bisectors are important in proving that a triangle is isosceles because they divide a line segment or angle into two equal parts. In an isosceles triangle, the bisector of an angle divides the opposite side into two equal parts, and the bisectors of two sides are equal. This relationship helps us prove that a triangle is isosceles.

Is it sufficient to show that only two sides of a triangle are equal to prove it is isosceles?

Yes, it is sufficient to show that two sides of a triangle are equal to prove it is isosceles. This is because the third side would automatically be equal to the other two sides in order for the triangle to be closed. Therefore, proving the equality of two sides is enough to prove that the triangle is isosceles.

Can the angle bisector theorem be used to prove that a triangle is isosceles?

Yes, the angle bisector theorem can be used to prove that a triangle is isosceles. This theorem states that in a triangle, the angle bisector of an angle divides the opposite side into two segments that are proportional to the other two sides of the triangle. If the two sides of the triangle are equal, then the two segments created by the angle bisector will also be equal, proving that the triangle is isosceles.

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