How far can a triangle center be from the triangle? [closed]
As is well known, a triangle center can be exterior to the triangle. So, just how far from the triangle can one of its centers be?
euclidean-geometry
asked Nov 27 '18 at 18:17
EulerSpoiler
729
closed as off-topic by Xander Henderson, Narasimham, Davide Giraudo, amWhy, KReiser Nov 28 '18 at 0:10
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Narasimham, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
As is well known, a triangle center can be exterior to the triangle. So, just how far from the triangle can one of its centers be?
euclidean-geometry
asked Nov 27 '18 at 18:17
EulerSpoiler
729
closed as off-topic by Xander Henderson, Narasimham, Davide Giraudo, amWhy, KReiser Nov 28 '18 at 0:10
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Narasimham, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
3
Do you mean the center of the circumcircle?
– Arthur
Nov 27 '18 at 18:19
3
If you scale such a triangle up, you can make it as far as you like ;)
– angryavian
Nov 27 '18 at 18:20
1
The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after?
– lulu
Nov 27 '18 at 18:23
add a comment |
As is well known, a triangle center can be exterior to the triangle. So, just how far from the triangle can one of its centers be?
euclidean-geometry
asked Nov 27 '18 at 18:17
EulerSpoiler
729
As is well known, a triangle center can be exterior to the triangle. So, just how far from the triangle can one of its centers be?
euclidean-geometry
euclidean-geometry
asked Nov 27 '18 at 18:17
EulerSpoiler
729
asked Nov 27 '18 at 18:17
EulerSpoiler
729
asked Nov 27 '18 at 18:17
EulerSpoiler
729
asked Nov 27 '18 at 18:17
EulerSpoiler
729
asked Nov 27 '18 at 18:17
EulerSpoiler
729
729
closed as off-topic by Xander Henderson, Narasimham, Davide Giraudo, amWhy, KReiser Nov 28 '18 at 0:10
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Narasimham, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Xander Henderson, Narasimham, Davide Giraudo, amWhy, KReiser Nov 28 '18 at 0:10
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Xander Henderson, Narasimham, Davide Giraudo
If this question can be reworded to fit the rules in the help center, please edit the question.
3
Do you mean the center of the circumcircle?
– Arthur
Nov 27 '18 at 18:19
3
If you scale such a triangle up, you can make it as far as you like ;)
– angryavian
Nov 27 '18 at 18:20
1
The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after?
– lulu
Nov 27 '18 at 18:23
add a comment |
3
Do you mean the center of the circumcircle?
– Arthur
Nov 27 '18 at 18:19
3
If you scale such a triangle up, you can make it as far as you like ;)
– angryavian
Nov 27 '18 at 18:20
1
The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after?
– lulu
Nov 27 '18 at 18:23
3
3
Do you mean the center of the circumcircle?
– Arthur
Nov 27 '18 at 18:19
Do you mean the center of the circumcircle?
– Arthur
Nov 27 '18 at 18:19
3
3
If you scale such a triangle up, you can make it as far as you like ;)
– angryavian
Nov 27 '18 at 18:20
If you scale such a triangle up, you can make it as far as you like ;)
– angryavian
Nov 27 '18 at 18:20
1
1
The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after?
– lulu
Nov 27 '18 at 18:23
The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after?
– lulu
Nov 27 '18 at 18:23
add a comment |
1 Answer
1
active
oldest
votes
As far as you want it to be. If you have a triangle whose circumcenter is 1 unit away from the triangle, you can just double all the sides of that triangle, and you get a triangle whose circumcenter is 2 units away. And so on.
Even if you limit yourself to, say, triangles with no sides longer than $1$ in order to make this scaling infeasible, you can do it another way. Consider if you have an isosceles triangle where two sides are $0.5$ and the angle between them is close to $180^circ$. The closer to $180^circ$ you make that angle (while keeping the triangle isosceles and the two side lengths constant), the further away the circumcenter moves.
And there is no bound on how far away the circumcenter can move in this manner; as our triangle comes closer and closer to being a degenerate triangle (three points on a line), the circumcircle comes closer and closer to being that line (which in this case is what a circle with infinite radius looks like).
answered Nov 27 '18 at 18:20
Arthur
110k7105186
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
As far as you want it to be. If you have a triangle whose circumcenter is 1 unit away from the triangle, you can just double all the sides of that triangle, and you get a triangle whose circumcenter is 2 units away. And so on.
Even if you limit yourself to, say, triangles with no sides longer than $1$ in order to make this scaling infeasible, you can do it another way. Consider if you have an isosceles triangle where two sides are $0.5$ and the angle between them is close to $180^circ$. The closer to $180^circ$ you make that angle (while keeping the triangle isosceles and the two side lengths constant), the further away the circumcenter moves.
And there is no bound on how far away the circumcenter can move in this manner; as our triangle comes closer and closer to being a degenerate triangle (three points on a line), the circumcircle comes closer and closer to being that line (which in this case is what a circle with infinite radius looks like).
answered Nov 27 '18 at 18:20
Arthur
110k7105186
add a comment |
As far as you want it to be. If you have a triangle whose circumcenter is 1 unit away from the triangle, you can just double all the sides of that triangle, and you get a triangle whose circumcenter is 2 units away. And so on.
Even if you limit yourself to, say, triangles with no sides longer than $1$ in order to make this scaling infeasible, you can do it another way. Consider if you have an isosceles triangle where two sides are $0.5$ and the angle between them is close to $180^circ$. The closer to $180^circ$ you make that angle (while keeping the triangle isosceles and the two side lengths constant), the further away the circumcenter moves.
And there is no bound on how far away the circumcenter can move in this manner; as our triangle comes closer and closer to being a degenerate triangle (three points on a line), the circumcircle comes closer and closer to being that line (which in this case is what a circle with infinite radius looks like).
answered Nov 27 '18 at 18:20
Arthur
110k7105186
add a comment |
As far as you want it to be. If you have a triangle whose circumcenter is 1 unit away from the triangle, you can just double all the sides of that triangle, and you get a triangle whose circumcenter is 2 units away. And so on.
Even if you limit yourself to, say, triangles with no sides longer than $1$ in order to make this scaling infeasible, you can do it another way. Consider if you have an isosceles triangle where two sides are $0.5$ and the angle between them is close to $180^circ$. The closer to $180^circ$ you make that angle (while keeping the triangle isosceles and the two side lengths constant), the further away the circumcenter moves.
And there is no bound on how far away the circumcenter can move in this manner; as our triangle comes closer and closer to being a degenerate triangle (three points on a line), the circumcircle comes closer and closer to being that line (which in this case is what a circle with infinite radius looks like).
answered Nov 27 '18 at 18:20
Arthur
110k7105186
As far as you want it to be. If you have a triangle whose circumcenter is 1 unit away from the triangle, you can just double all the sides of that triangle, and you get a triangle whose circumcenter is 2 units away. And so on.
Even if you limit yourself to, say, triangles with no sides longer than $1$ in order to make this scaling infeasible, you can do it another way. Consider if you have an isosceles triangle where two sides are $0.5$ and the angle between them is close to $180^circ$. The closer to $180^circ$ you make that angle (while keeping the triangle isosceles and the two side lengths constant), the further away the circumcenter moves.
And there is no bound on how far away the circumcenter can move in this manner; as our triangle comes closer and closer to being a degenerate triangle (three points on a line), the circumcircle comes closer and closer to being that line (which in this case is what a circle with infinite radius looks like).
answered Nov 27 '18 at 18:20
Arthur
110k7105186
edited Nov 27 '18 at 18:25
answered Nov 27 '18 at 18:20
Arthur
110k7105186
answered Nov 27 '18 at 18:20
Arthur
110k7105186
answered Nov 27 '18 at 18:20
Arthur
110k7105186
110k7105186
add a comment |
add a comment |
3
Do you mean the center of the circumcircle?
– Arthur
Nov 27 '18 at 18:19
3
If you scale such a triangle up, you can make it as far as you like ;)
– angryavian
Nov 27 '18 at 18:20
1
The question isn't clear. How are you defining the distance? Euler tells us that the distance between the incenter and the circumcenter satisfies $d^2=R(R-2r)$. Is that the sort of thing you are after?
– lulu
Nov 27 '18 at 18:23