Prove that the longest line segment in a triangle is on the sides.

[yma16]

Prove that the longest line segment in a triangle is on the sides.

 

[yma16]

Please post something that makes sense...

If it does not make sense, please show a counterexample.

 

[yma16]

Read my post again: edited it!

What I can do is to show that the end points of the line segment are on the sides. Because for any end point in the triangle that is not on the side, we can extend the the line to make it to cross the side. The new end point will make the line segment longer.

 

[Steven G]

is on the sides.

I'll take a crack at this.
Given a triangle there are three line segments and all these line segments make up the three sides of the triangle. The sides all have a length. Since there a finite number of of sides (3 to be exact)we can find the length of the largest side. As all the line segments are on the triangle it follows that the longest line segment is on the triangle. QED.

 

[Dr.Peterson]

I'll take a crack at this.
Given a triangle there are three line segments and all these line segments make up the three sides of the triangle. The sides all have a length. Since there a finite number of of sides (3 to be exact)we can find the length of the largest side. As all the line segments are on the triangle it follows that the longest line segment is on the triangle. QED.

My first concern with the problem is in defining what it means:

Prove that the longest line segment in a triangle is on the sides.
​

What does it mean for a segment to be in a triangle? And is "on the sides" really what was intended? We can't really talk about a proof without clarifying this.

Jomo, you have evidently taken "in a triangle" to mean "on a side". That, of course, makes the problem almost vacuous.

On the other hand, in my mind (technically) "in a triangle" should mean in its interior, which the sides are not. So the statement has to be false in this sense. (Sometimes "triangle" or "polygon" is used to refer only to the union of the edges, as "circle" refers only to its circumference; other times these words are taken to refer to the entire figure including its interior.)

So I propose that the problem be restated as

Prove that the longest line segment contained in the union of a triangle with its interior is one of its edges.
​

yma16, is that your intention? I agree with your assessment that a truly careful proof is tricky, and will probably take several steps, starting with what you showed.

 

[yma16]

I agree with you. The triangle is a closed and solid figure.

My first concern with the problem is in defining what it means:

Prove that the longest line segment in a triangle is on the sides.
​

What does it mean for a segment to be in a triangle? And is "on the sides" really what was intended? We can't really talk about a proof without clarifying this.

Jomo, you have evidently taken "in a triangle" to mean "on a side". That, of course, makes the problem almost vacuous.

On the other hand, in my mind (technically) "in a triangle" should mean in its interior, which the sides are not. So the statement has to be false in this sense. (Sometimes "triangle" or "polygon" is used to refer only to the union of the edges, as "circle" refers only to its circumference; other times these words are taken to refer to the entire figure including its interior.)

So I propose that the problem be restated as

Prove that the longest line segment contained in the union of a triangle with its interior is one of its edges.
​

yma16, is that your intention? I agree with your assessment that a truly careful proof is tricky, and will probably take several steps, starting with what you showed.

I agree with you. The triangle is a closed and solid figure. It includes the interior and three edges. The step 1 is to show the end points should be on the edge(s), which I have done. My step 2 is to show that one of the end point should be a vortex. The step 3 is to complete the proof. I am working on step 2 now.

Thank you for making it more clear.

 

[yma16]

Step 3 is done.

It includes the interior and three edges. The step 1 is to show the end points should be on the edge(s), which I have done. My step 2 is to show that one of the end point should be a vortex. The step 3 is to complete the proof. I am working on step 2 now.

Thank you for making it more clear.

Step 3 is done.

assume one endpoint is a vortex and the other endpoint on the opposite edge (not the vortex). This line create two small triangles. Then, the line segment and the opposite edge form two angles. One of the angles is either 90 degree or bigger than 90 degree. If the angle is 90 degree, then the line segment becomes a leg of the small triangle, hence it is not the longest line segment in the original triangle. The second case is the angle is bigger than 90 degree. Then this angle's opposite edge is longer than the line segment, since larger angle's opposite edge is bigger than smaller angle's opposite edge and within a triangle, there is at most one angle that can be bigger than 90 degree.

The conclusion is, the line segment with only one endpoint being a vortex is not the longest in a triangle.

I am still working step 2.

 

[yma16]

step 2 is similar to step 3

assume one endpoint is a vortex and the other endpoint on the opposite edge (not the vortex). This line create two small triangles. Then, the line segment and the opposite edge form two angles. One of the angles is either 90 degree or bigger than 90 degree. If the angle is 90 degree, then the line segment becomes a leg of the small triangle, hence it is not the longest line segment in the original triangle. The second case is the angle is bigger than 90 degree. Then this angle's opposite edge is longer than the line segment, since larger angle's opposite edge is bigger than smaller angle's opposite edge and within a triangle, there is at most one angle that can be bigger than 90 degree.

The conclusion is, the line segment with only one endpoint being a vortex is not the longest in a triangle.

I am still working step 2.

step 2 is similar to step 3




I am going to prove DE is not the longest line in triangle ABC.

If <ADE = 90 degree, then DE is a leg in the right triangle ADE. Hence mDE<mAE.
If <ADE > 90 degree, then <ADE > <DAE. Hence mDE<mAE.
If <ADE < 90 degree, then <BDE>90 degree. Therefore <DBE < 90 degree. Hence mDE < mBE.

So DE is not the longest line in the triangle ABC.

There is a trivial case such that the line segment is on a edge and it is <=the edge.

 

[Dr.Peterson]

View attachment 9365

I am going to prove DE is not the longest line in triangle ABC.

If <ADE = 90 degree, then DE is a leg in the right triangle ADE. Hence mDE<mAE.
If <ADE > 90 degree, then <ADE > <DAE. Hence mDE<mAE.
If <ADE < 90 degree, then <BDE>90 degree. Therefore <DBE < 90 degree. Hence mDE < mBE.

So DE is not the longest line in the triangle ABC.

There is a trivial case such that the line segment is on a edge and it is <=the edge.

You're doing very nicely. As I think you're seeing, the proof is not really hard, just unwieldy because of all the cases to consider. Your step 3 can possibly be tightened up a little bit (by writing more like you wrote this one), but I think you are done (for a first draft).