Three sides of a regular triangle is bicolored, are there three points with the same color forming a...
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The sides of a regular triangle $triangle_1=ABC$ is bicolored(red, and blue), Do there exist three vertices on the perimeter of $triangle_1$ three monochromatic vertices forming the corners of a rectangular triangle?
So here is my attempt: Two of the tree vertices, say $A,B$ of $triangle_1$ must be colored the same, say blue, by pigeonhole principle.
Then if no point, excluding $A$ and $B$, on the edge $AC$ and $BC$ are colored blue, then we can find three red vertices on edges $AC$ and $BC$ to form a rectangular triangle. Otherwise there exist a point $D$ on $AC$ or $BC$ colored blue. But How can we make it into a rectangular triangle?
Also, is there a good drawing software for math?
I found a solution here but I don't understand it:
Suppose there is no right triangle with vertices of the same color. Partition each side of the regular triangle by two points into three equal parts. These points are vertices of a regular hexagon. If two of its opposite vertices are of the same color, then all other vertices are of the other color, and hence there exists a right triangle with vertices of the other color. Hence opposite vertices of the hexagon are of different color. Thus there exist two neighboring vertices of different color. One pair of these bicolored vertices lies on a side of the triangle. The points of this side, differing from the vertices of the hexagon, cannot be of the first or second color. Contradiction.
My question is why the last sentence :"The points of this side, differing from the vertices of the hexagon, cannot be of the first or the second color" true?
geometry contest-math pigeonhole-principle
asked Nov 17 at 0:14
mathnoob
1,352116
add a comment |
up vote
1
down vote
favorite
The sides of a regular triangle $triangle_1=ABC$ is bicolored(red, and blue), Do there exist three vertices on the perimeter of $triangle_1$ three monochromatic vertices forming the corners of a rectangular triangle?
So here is my attempt: Two of the tree vertices, say $A,B$ of $triangle_1$ must be colored the same, say blue, by pigeonhole principle.
Then if no point, excluding $A$ and $B$, on the edge $AC$ and $BC$ are colored blue, then we can find three red vertices on edges $AC$ and $BC$ to form a rectangular triangle. Otherwise there exist a point $D$ on $AC$ or $BC$ colored blue. But How can we make it into a rectangular triangle?
Also, is there a good drawing software for math?
I found a solution here but I don't understand it:
Suppose there is no right triangle with vertices of the same color. Partition each side of the regular triangle by two points into three equal parts. These points are vertices of a regular hexagon. If two of its opposite vertices are of the same color, then all other vertices are of the other color, and hence there exists a right triangle with vertices of the other color. Hence opposite vertices of the hexagon are of different color. Thus there exist two neighboring vertices of different color. One pair of these bicolored vertices lies on a side of the triangle. The points of this side, differing from the vertices of the hexagon, cannot be of the first or second color. Contradiction.
My question is why the last sentence :"The points of this side, differing from the vertices of the hexagon, cannot be of the first or the second color" true?
geometry contest-math pigeonhole-principle
asked Nov 17 at 0:14
mathnoob
1,352116
By rectangular, do you mean right-angled? I.e. A triangle in which one of the angles is a right angle?
– астон вілла олоф мэллбэрг
Nov 17 at 3:50
A good piece of software for drawing math
– Quang Hoang
Nov 17 at 5:16
yes, rectangle triangle means right angled triangle.
– mathnoob
Nov 17 at 9:22
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The sides of a regular triangle $triangle_1=ABC$ is bicolored(red, and blue), Do there exist three vertices on the perimeter of $triangle_1$ three monochromatic vertices forming the corners of a rectangular triangle?
So here is my attempt: Two of the tree vertices, say $A,B$ of $triangle_1$ must be colored the same, say blue, by pigeonhole principle.
Then if no point, excluding $A$ and $B$, on the edge $AC$ and $BC$ are colored blue, then we can find three red vertices on edges $AC$ and $BC$ to form a rectangular triangle. Otherwise there exist a point $D$ on $AC$ or $BC$ colored blue. But How can we make it into a rectangular triangle?
Also, is there a good drawing software for math?
I found a solution here but I don't understand it:
Suppose there is no right triangle with vertices of the same color. Partition each side of the regular triangle by two points into three equal parts. These points are vertices of a regular hexagon. If two of its opposite vertices are of the same color, then all other vertices are of the other color, and hence there exists a right triangle with vertices of the other color. Hence opposite vertices of the hexagon are of different color. Thus there exist two neighboring vertices of different color. One pair of these bicolored vertices lies on a side of the triangle. The points of this side, differing from the vertices of the hexagon, cannot be of the first or second color. Contradiction.
My question is why the last sentence :"The points of this side, differing from the vertices of the hexagon, cannot be of the first or the second color" true?
geometry contest-math pigeonhole-principle
asked Nov 17 at 0:14
mathnoob
1,352116
The sides of a regular triangle $triangle_1=ABC$ is bicolored(red, and blue), Do there exist three vertices on the perimeter of $triangle_1$ three monochromatic vertices forming the corners of a rectangular triangle?
So here is my attempt: Two of the tree vertices, say $A,B$ of $triangle_1$ must be colored the same, say blue, by pigeonhole principle.
Then if no point, excluding $A$ and $B$, on the edge $AC$ and $BC$ are colored blue, then we can find three red vertices on edges $AC$ and $BC$ to form a rectangular triangle. Otherwise there exist a point $D$ on $AC$ or $BC$ colored blue. But How can we make it into a rectangular triangle?
Also, is there a good drawing software for math?
I found a solution here but I don't understand it:
Suppose there is no right triangle with vertices of the same color. Partition each side of the regular triangle by two points into three equal parts. These points are vertices of a regular hexagon. If two of its opposite vertices are of the same color, then all other vertices are of the other color, and hence there exists a right triangle with vertices of the other color. Hence opposite vertices of the hexagon are of different color. Thus there exist two neighboring vertices of different color. One pair of these bicolored vertices lies on a side of the triangle. The points of this side, differing from the vertices of the hexagon, cannot be of the first or second color. Contradiction.
My question is why the last sentence :"The points of this side, differing from the vertices of the hexagon, cannot be of the first or the second color" true?
geometry contest-math pigeonhole-principle
geometry contest-math pigeonhole-principle
asked Nov 17 at 0:14
mathnoob
1,352116
asked Nov 17 at 0:14
mathnoob
1,352116
edited Nov 17 at 9:56
asked Nov 17 at 0:14
mathnoob
1,352116
asked Nov 17 at 0:14
mathnoob
1,352116
asked Nov 17 at 0:14
mathnoob
1,352116
1,352116
By rectangular, do you mean right-angled? I.e. A triangle in which one of the angles is a right angle?
– астон вілла олоф мэллбэрг
Nov 17 at 3:50
A good piece of software for drawing math
– Quang Hoang
Nov 17 at 5:16
yes, rectangle triangle means right angled triangle.
– mathnoob
Nov 17 at 9:22
add a comment |
By rectangular, do you mean right-angled? I.e. A triangle in which one of the angles is a right angle?
– астон вілла олоф мэллбэрг
Nov 17 at 3:50
A good piece of software for drawing math
– Quang Hoang
Nov 17 at 5:16
yes, rectangle triangle means right angled triangle.
– mathnoob
Nov 17 at 9:22
By rectangular, do you mean right-angled? I.e. A triangle in which one of the angles is a right angle?
– астон вілла олоф мэллбэрг
Nov 17 at 3:50
By rectangular, do you mean right-angled? I.e. A triangle in which one of the angles is a right angle?
– астон вілла олоф мэллбэрг
Nov 17 at 3:50
A good piece of software for drawing math
– Quang Hoang
Nov 17 at 5:16
A good piece of software for drawing math
– Quang Hoang
Nov 17 at 5:16
yes, rectangle triangle means right angled triangle.
– mathnoob
Nov 17 at 9:22
yes, rectangle triangle means right angled triangle.
– mathnoob
Nov 17 at 9:22
add a comment |
1 Answer
1
active
oldest
votes
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0
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accepted
(Note: A first version of this answer was given before your edit.) I claim that a monochromatic right triangle cannot be avoided.
Proof. The smaller triangle $triangle(DEF)$ in the following figure has at least two vertices of equal color. We may assume that $D$ and $E$ are red. This enforces $P$ and $Q$ both to be blue, or there would be a monochromatic red triangle. Now the point $C$ can be colored neither red nor blue without creating a monochromatic right triangle.
answered Nov 17 at 10:41
Christian Blatter
171k7111325
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
(Note: A first version of this answer was given before your edit.) I claim that a monochromatic right triangle cannot be avoided.
Proof. The smaller triangle $triangle(DEF)$ in the following figure has at least two vertices of equal color. We may assume that $D$ and $E$ are red. This enforces $P$ and $Q$ both to be blue, or there would be a monochromatic red triangle. Now the point $C$ can be colored neither red nor blue without creating a monochromatic right triangle.
answered Nov 17 at 10:41
Christian Blatter
171k7111325
add a comment |
up vote
0
down vote
accepted
(Note: A first version of this answer was given before your edit.) I claim that a monochromatic right triangle cannot be avoided.
Proof. The smaller triangle $triangle(DEF)$ in the following figure has at least two vertices of equal color. We may assume that $D$ and $E$ are red. This enforces $P$ and $Q$ both to be blue, or there would be a monochromatic red triangle. Now the point $C$ can be colored neither red nor blue without creating a monochromatic right triangle.
answered Nov 17 at 10:41
Christian Blatter
171k7111325
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
(Note: A first version of this answer was given before your edit.) I claim that a monochromatic right triangle cannot be avoided.
Proof. The smaller triangle $triangle(DEF)$ in the following figure has at least two vertices of equal color. We may assume that $D$ and $E$ are red. This enforces $P$ and $Q$ both to be blue, or there would be a monochromatic red triangle. Now the point $C$ can be colored neither red nor blue without creating a monochromatic right triangle.
answered Nov 17 at 10:41
Christian Blatter
171k7111325
(Note: A first version of this answer was given before your edit.) I claim that a monochromatic right triangle cannot be avoided.
Proof. The smaller triangle $triangle(DEF)$ in the following figure has at least two vertices of equal color. We may assume that $D$ and $E$ are red. This enforces $P$ and $Q$ both to be blue, or there would be a monochromatic red triangle. Now the point $C$ can be colored neither red nor blue without creating a monochromatic right triangle.
answered Nov 17 at 10:41
Christian Blatter
171k7111325
edited Nov 17 at 16:56
answered Nov 17 at 10:41
Christian Blatter
171k7111325
answered Nov 17 at 10:41
Christian Blatter
171k7111325
answered Nov 17 at 10:41
Christian Blatter
171k7111325
171k7111325
add a comment |
add a comment |
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By rectangular, do you mean right-angled? I.e. A triangle in which one of the angles is a right angle?
– астон вілла олоф мэллбэрг
Nov 17 at 3:50
A good piece of software for drawing math
– Quang Hoang
Nov 17 at 5:16
yes, rectangle triangle means right angled triangle.
– mathnoob
Nov 17 at 9:22