Feuerbach's Theorem in Spherical Triangle?
1
So first of all, just incase you don't know, Feuerbach's Theorem states the nine-point circle of a triangle in Euclidean space is tangent to the three excircles and the incircle of the triangle. However, does this theorem hold for spherical triangles? Because it seems like you can't have tangent sides of a triangle to a circle on a spherical surface? If so, can another theorem be found with a similar relation besides tangency?
triangle spherical-geometry
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
add a comment |
1
So first of all, just incase you don't know, Feuerbach's Theorem states the nine-point circle of a triangle in Euclidean space is tangent to the three excircles and the incircle of the triangle. However, does this theorem hold for spherical triangles? Because it seems like you can't have tangent sides of a triangle to a circle on a spherical surface? If so, can another theorem be found with a similar relation besides tangency?
triangle spherical-geometry
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
Feuerbach aside, you seem to have a more-fundamental question: Does a spherical triangle have in- and ex-circles? The answer is: Yes. (I trust you know that the "lines" on a sphere are great circles.) It's pretty easy to convince oneself that an incircle exists: Close to a corner of a triangle, one can certainly draw a circle tangent to the two sides; from here, all one needs to do is make the circle larger and larger, maintaining tangency all the while, until it bumps into the opposite side. Voilà! Incircle! (Uniqueness takes a little more work.) Similarly for excircles.
– Blue
Nov 30 '18 at 11:12
Thank you. I did know that lines form circles on spheres. That's why the angles are 90 degrees in triangles. The lines from the top to bottom(the "longitudes") and the sideways lines(the "latitudes") are always perpendicular. So if you take three segments of three lines to form a spherical triangle, that rule will apply as long as the lines are longitude and latitude-like. For lines not perfectly vertical or horizontal, there is a bit of a difference, but that's not what I was after
– Xavier Stanton
Nov 30 '18 at 16:08
1
"That's why the angles are 90 degrees in triangles." Ummm ... no. Lines on the sphere aren't just circles, they're great circles formed by cutting the sphere with a plane through its center. (Not-so-great circles aren't spherical lines, they're just circles.) So, the line between two points is determined by the plane through those points and the center. The angle between two spherical lines (ie, great circles) is defined to be the angle between their associated planes; so, angles can have any size. Interestingly, the angle-sum in a spherical triangle is always more than $180^circ$.
– Blue
Nov 30 '18 at 17:01
Whoops, my mistake. Thank you for clearing that up.
– Xavier Stanton
Nov 30 '18 at 17:21
add a comment |
1
1
1
So first of all, just incase you don't know, Feuerbach's Theorem states the nine-point circle of a triangle in Euclidean space is tangent to the three excircles and the incircle of the triangle. However, does this theorem hold for spherical triangles? Because it seems like you can't have tangent sides of a triangle to a circle on a spherical surface? If so, can another theorem be found with a similar relation besides tangency?
triangle spherical-geometry
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
So first of all, just incase you don't know, Feuerbach's Theorem states the nine-point circle of a triangle in Euclidean space is tangent to the three excircles and the incircle of the triangle. However, does this theorem hold for spherical triangles? Because it seems like you can't have tangent sides of a triangle to a circle on a spherical surface? If so, can another theorem be found with a similar relation besides tangency?
triangle spherical-geometry
triangle spherical-geometry
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
asked Nov 29 '18 at 20:42
Xavier StantonXavier Stanton
309211
309211
Feuerbach aside, you seem to have a more-fundamental question: Does a spherical triangle have in- and ex-circles? The answer is: Yes. (I trust you know that the "lines" on a sphere are great circles.) It's pretty easy to convince oneself that an incircle exists: Close to a corner of a triangle, one can certainly draw a circle tangent to the two sides; from here, all one needs to do is make the circle larger and larger, maintaining tangency all the while, until it bumps into the opposite side. Voilà! Incircle! (Uniqueness takes a little more work.) Similarly for excircles.
– Blue
Nov 30 '18 at 11:12
Thank you. I did know that lines form circles on spheres. That's why the angles are 90 degrees in triangles. The lines from the top to bottom(the "longitudes") and the sideways lines(the "latitudes") are always perpendicular. So if you take three segments of three lines to form a spherical triangle, that rule will apply as long as the lines are longitude and latitude-like. For lines not perfectly vertical or horizontal, there is a bit of a difference, but that's not what I was after
– Xavier Stanton
Nov 30 '18 at 16:08
1
"That's why the angles are 90 degrees in triangles." Ummm ... no. Lines on the sphere aren't just circles, they're great circles formed by cutting the sphere with a plane through its center. (Not-so-great circles aren't spherical lines, they're just circles.) So, the line between two points is determined by the plane through those points and the center. The angle between two spherical lines (ie, great circles) is defined to be the angle between their associated planes; so, angles can have any size. Interestingly, the angle-sum in a spherical triangle is always more than $180^circ$.
– Blue
Nov 30 '18 at 17:01
Whoops, my mistake. Thank you for clearing that up.
– Xavier Stanton
Nov 30 '18 at 17:21
add a comment |
Feuerbach aside, you seem to have a more-fundamental question: Does a spherical triangle have in- and ex-circles? The answer is: Yes. (I trust you know that the "lines" on a sphere are great circles.) It's pretty easy to convince oneself that an incircle exists: Close to a corner of a triangle, one can certainly draw a circle tangent to the two sides; from here, all one needs to do is make the circle larger and larger, maintaining tangency all the while, until it bumps into the opposite side. Voilà! Incircle! (Uniqueness takes a little more work.) Similarly for excircles.
– Blue
Nov 30 '18 at 11:12
Thank you. I did know that lines form circles on spheres. That's why the angles are 90 degrees in triangles. The lines from the top to bottom(the "longitudes") and the sideways lines(the "latitudes") are always perpendicular. So if you take three segments of three lines to form a spherical triangle, that rule will apply as long as the lines are longitude and latitude-like. For lines not perfectly vertical or horizontal, there is a bit of a difference, but that's not what I was after
– Xavier Stanton
Nov 30 '18 at 16:08
1
"That's why the angles are 90 degrees in triangles." Ummm ... no. Lines on the sphere aren't just circles, they're great circles formed by cutting the sphere with a plane through its center. (Not-so-great circles aren't spherical lines, they're just circles.) So, the line between two points is determined by the plane through those points and the center. The angle between two spherical lines (ie, great circles) is defined to be the angle between their associated planes; so, angles can have any size. Interestingly, the angle-sum in a spherical triangle is always more than $180^circ$.
– Blue
Nov 30 '18 at 17:01
Whoops, my mistake. Thank you for clearing that up.
– Xavier Stanton
Nov 30 '18 at 17:21
Feuerbach aside, you seem to have a more-fundamental question: Does a spherical triangle have in- and ex-circles? The answer is: Yes. (I trust you know that the "lines" on a sphere are great circles.) It's pretty easy to convince oneself that an incircle exists: Close to a corner of a triangle, one can certainly draw a circle tangent to the two sides; from here, all one needs to do is make the circle larger and larger, maintaining tangency all the while, until it bumps into the opposite side. Voilà! Incircle! (Uniqueness takes a little more work.) Similarly for excircles.
– Blue
Nov 30 '18 at 11:12
Feuerbach aside, you seem to have a more-fundamental question: Does a spherical triangle have in- and ex-circles? The answer is: Yes. (I trust you know that the "lines" on a sphere are great circles.) It's pretty easy to convince oneself that an incircle exists: Close to a corner of a triangle, one can certainly draw a circle tangent to the two sides; from here, all one needs to do is make the circle larger and larger, maintaining tangency all the while, until it bumps into the opposite side. Voilà! Incircle! (Uniqueness takes a little more work.) Similarly for excircles.
– Blue
Nov 30 '18 at 11:12
Thank you. I did know that lines form circles on spheres. That's why the angles are 90 degrees in triangles. The lines from the top to bottom(the "longitudes") and the sideways lines(the "latitudes") are always perpendicular. So if you take three segments of three lines to form a spherical triangle, that rule will apply as long as the lines are longitude and latitude-like. For lines not perfectly vertical or horizontal, there is a bit of a difference, but that's not what I was after
– Xavier Stanton
Nov 30 '18 at 16:08
Thank you. I did know that lines form circles on spheres. That's why the angles are 90 degrees in triangles. The lines from the top to bottom(the "longitudes") and the sideways lines(the "latitudes") are always perpendicular. So if you take three segments of three lines to form a spherical triangle, that rule will apply as long as the lines are longitude and latitude-like. For lines not perfectly vertical or horizontal, there is a bit of a difference, but that's not what I was after
– Xavier Stanton
Nov 30 '18 at 16:08
1
1
"That's why the angles are 90 degrees in triangles." Ummm ... no. Lines on the sphere aren't just circles, they're great circles formed by cutting the sphere with a plane through its center. (Not-so-great circles aren't spherical lines, they're just circles.) So, the line between two points is determined by the plane through those points and the center. The angle between two spherical lines (ie, great circles) is defined to be the angle between their associated planes; so, angles can have any size. Interestingly, the angle-sum in a spherical triangle is always more than $180^circ$.
– Blue
Nov 30 '18 at 17:01
"That's why the angles are 90 degrees in triangles." Ummm ... no. Lines on the sphere aren't just circles, they're great circles formed by cutting the sphere with a plane through its center. (Not-so-great circles aren't spherical lines, they're just circles.) So, the line between two points is determined by the plane through those points and the center. The angle between two spherical lines (ie, great circles) is defined to be the angle between their associated planes; so, angles can have any size. Interestingly, the angle-sum in a spherical triangle is always more than $180^circ$.
– Blue
Nov 30 '18 at 17:01
Whoops, my mistake. Thank you for clearing that up.
– Xavier Stanton
Nov 30 '18 at 17:21
Whoops, my mistake. Thank you for clearing that up.
– Xavier Stanton
Nov 30 '18 at 17:21
add a comment |
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Feuerbach aside, you seem to have a more-fundamental question: Does a spherical triangle have in- and ex-circles? The answer is: Yes. (I trust you know that the "lines" on a sphere are great circles.) It's pretty easy to convince oneself that an incircle exists: Close to a corner of a triangle, one can certainly draw a circle tangent to the two sides; from here, all one needs to do is make the circle larger and larger, maintaining tangency all the while, until it bumps into the opposite side. Voilà! Incircle! (Uniqueness takes a little more work.) Similarly for excircles.
– Blue
Nov 30 '18 at 11:12
Thank you. I did know that lines form circles on spheres. That's why the angles are 90 degrees in triangles. The lines from the top to bottom(the "longitudes") and the sideways lines(the "latitudes") are always perpendicular. So if you take three segments of three lines to form a spherical triangle, that rule will apply as long as the lines are longitude and latitude-like. For lines not perfectly vertical or horizontal, there is a bit of a difference, but that's not what I was after
– Xavier Stanton
Nov 30 '18 at 16:08
1
"That's why the angles are 90 degrees in triangles." Ummm ... no. Lines on the sphere aren't just circles, they're great circles formed by cutting the sphere with a plane through its center. (Not-so-great circles aren't spherical lines, they're just circles.) So, the line between two points is determined by the plane through those points and the center. The angle between two spherical lines (ie, great circles) is defined to be the angle between their associated planes; so, angles can have any size. Interestingly, the angle-sum in a spherical triangle is always more than $180^circ$.
– Blue
Nov 30 '18 at 17:01
Whoops, my mistake. Thank you for clearing that up.
– Xavier Stanton
Nov 30 '18 at 17:21