Proving At Least 3 Pure Triangles in a Complete Heptagon

In summary, to prove that when edges of a complete heptagon are colored with two different colors, there will be at least three pure triangles, we first consider a vertex v with 6 edges incident to it. At least 3 of these edges will be the same color, giving us 2 pure triangles. If at least 4 edges are the same color, we can have at most 2 pure triangles. However, if we consider the outside edges r-s-t-u, we see that they must be different colors in order to avoid having 3 pure triangles. This ultimately leads to the creation of a third pure triangle.
  • #1
Solarmew
37
1
errrr, THEOREM >.< ... oops ...

Homework Statement


Prove that when edges of a complete heptagon are colored with two different colors, there will be at least three pure triangles.

Homework Equations


The Attempt at a Solution


i can do two pure triangles, but not three :cry:
pick a vertex v. It has 6 edges incident to it, at least 3 of which are the same color.
1. Suppose 3 of these edges, connecting to vertices r, s and t, are blue. If any of the edges (r, s), (r, t), (s, t) are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle. Same goes for the other 3 edges incident to v, say red, that are connected to other 3 vertices, say a, b, c. Therefore we have 2 pure triangles.
2. Suppose at least 4 edges connecting v to r, s, t, u are blue. Then the least amount of pure triangles connecting r,s,t,u is 0 when the internal diagonals are blue and the outside square is red. But since we have two blue diagonals, each connected to v by two blue edges, we have 2 pure triangles.

now any ideas as far as how to get the third? :confused:
 
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  • #2
i've been fiddling with the picture in photoshop XD because I'm a visual learner XD
1.JPG

and i kinda see what's going on, just not sure how to put it into words, or if I'm even on the right track or if I'm over complicating this X.X

but if there are at least 4 edges incident to v s.t. we have at most 2 pure triangles (if we connect the other two red vertices with the same color line, we're done, otherwise...), then the outsides edges r-s-t-u are all red, which means if they make up the bases of some triangles, the other two edges for each of those triangles must be different colors (since we want to try to avoid having 3 pure triangles, but seeing if we'll be forced to)
2.JPG
3.JPG


since we have to alternate colors, they eventually create another triangle ... so I'm convinced of that now ... just don't know how to show it properly :\

i can also kinda see how we're forced to have the third pure triangle when we have 3 red and 3 blue edges coming out of v, but again, not sure how to put it into math form XD
 
Last edited:
  • #3
No one? XD well at least I dun feel so bad now XD
 

1. How do you define a pure triangle in a complete heptagon?

A pure triangle in a complete heptagon is a triangle formed by three vertices that are also vertices of the heptagon, and all three sides of the triangle are line segments of the heptagon.

2. What is a complete heptagon?

A complete heptagon is a polygon with seven sides and seven angles, where all sides and angles are equal in length and measure, respectively.

3. Can a complete heptagon have more than 3 pure triangles?

Yes, a complete heptagon can have up to 21 pure triangles. This is because each vertex of the heptagon can be used as the starting point for three different pure triangles.

4. How can you prove that a complete heptagon has at least 3 pure triangles?

To prove that a complete heptagon has at least 3 pure triangles, you can use the fact that the sum of the interior angles of any polygon is equal to (n-2)180 degrees, where n is the number of sides. In this case, n=7, so the sum of the interior angles is 5*180= 900 degrees. Since all angles in a complete heptagon are equal, each angle must measure 900/7 = 128.57 degrees. By constructing three triangles with these angles, we can prove that at least 3 pure triangles exist in a complete heptagon.

5. Are there any other methods to prove the existence of at least 3 pure triangles in a complete heptagon?

Yes, there are other methods to prove the existence of at least 3 pure triangles in a complete heptagon. One method is to use symmetry, where we can divide the heptagon into smaller triangles and use symmetry to show that at least three of these smaller triangles are pure. Another method is to use the properties of a regular heptagon, which has rotational symmetry of order 7, to prove that there must be at least 3 pure triangles in a complete heptagon.

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