Triangle, trisected

[bonanova]

Warm-up problem:

Bisect the angles of a triangle.

Describe the point(s) where each bisector first intersects one of the others.

Now try this:

Trisect the angles of a triangle.

Describe the points where each trisector first intersects one of the others.

[kukupai]

... this triangle is equilateral.

(If my drawing was correct.)

[Debasis]

Do we have to name that point?

if we have to name then the angle bisectors of a triangle meet at the incentre (i learned about it in school) and i dont know about the point where trisection meet.....i do hope you want us to find coordinates of the point...

[bonanova]

Do we have to name that point?

if we have to name then the angle bisectors of a triangle meet at the incentre (i learned about it in school) and i dont know about the point where trisection meet.....i do hope you want us to find coordinates of the point...

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.

And there is something special about those three points.

[Debasis]

Do we have to name that point?

if we have to name then the angle bisectors of a triangle meet at the incentre (i learned about it in school) and i dont know about the point where trisection meet.....i do hope you want us to find coordinates of the point...

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.

And there is something special about those three points.

the three point where the trisectors meet lie on the same line...

[bonanova]

Do we have to name that point?

if we have to name then the angle bisectors of a triangle meet at the incentre (i learned about it in school) and i dont know about the point where trisection meet.....i do hope you want us to find coordinates of the point...

You're right about the bisector case.

But for the trisector case there is more than one point.

In fact there are three places where a trisector of one angle first intersects a trisector of one of the other angles.

And there is something special about those three points.

the three point where the trisectors meet lie on the same line...

Actually they are not collinear.

That being the case, they form a triangle.

So the OP really asks: what is special about the triangle formed by these three points?

[bonanova]

... this triangle is equilateral.(If my drawing was correct.)

Correct. Good job.

I wonder if there is a proof of this that is not overly complex?

Edit:

Well, No. I just found the proof, and it's not beautiful for its simplicity.

You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.

"Do not try this at home."

Edited March 15, 2013 by bonanova
Comment on proof

[Debasis]

... this triangle is equilateral.(If my drawing was correct.)

Correct. Good job.

I wonder if there is a proof of this that is not overly complex?

Edit:

Well, No. I just found the proof, and it's not beautiful for its simplicity.

You start with the Law of Sines, and 2 1/2 pages later you have a symmetrical expression for one side.

"Do not try this at home."

oh i am so sorry .....i mistook it for the eulers line....my bad...

[marksmanjay]

COMBINATION: 7129