Three externally tangent circles sharing a single point of tangency
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Three externally tangent circles sharing a single point of tangency
I would like to prove that it is impossible for three externally tangent circles in the same plane to have a single point of tangency.
In other words, if you have two circles (A & B) that are externally tangent to each other with point of tangency X, can there be a third circle (C) that is externally tangent to A and B, and tangent to those circles with the same point of tangency X?
It seems obvious that it is true because if I try to draw three circles like that it seems impossible, but I would like a better proof than that.
Any advice on how I should proceed? My geometry is extremely rusty and I don't know how to begin.
In other words, if you have two circles (A & B) that are externally tangent to each other with point of tangency X, can there be a third circle (C) that is externally tangent to A and B, and tangent to those circles with the same point of tangency X?
It seems obvious that it is true because if I try to draw three circles like that it seems impossible, but I would like a better proof than that.
Any advice on how I should proceed? My geometry is extremely rusty and I don't know how to begin.
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Re: Three externally tangent circles sharing a single point of tangency
I don't have a clear proof in mind, but I would probably start with the angle made between the tangent line and the radii connecting the circle centers to the tangent point.
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Re: Three externally tangent circles sharing a single point of tangency
To be clear, all three circles are coplanar? Because otherwise it definitely is not true.

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Re: Three externally tangent circles sharing a single point of tangency
Eebster the Great wrote:To be clear, all three circles are coplanar? Because otherwise it definitely is not true.
Coplanar circles, yes!
Zohar wrote:I don't have a clear proof in mind, but I would probably start with the angle made between the tangent line and the radii connecting the circle centers to the tangent point.
I think this should help! If I can prove that the radius of a circle is perpendicular to a tangent line, I should be able to just add up the size of the angles and prove that they add up to more than 360 degrees. Let me mull on this.
Nice! Thank you.
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Re: Three externally tangent circles sharing a single point of tangency
I mean, that's true  the angle between a tangent and radius is always 90 degrees (in the usual 2D geometry we're talking about).
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Re: Three externally tangent circles sharing a single point of tangency
One less thing to worry about then! Thanks.
I've been getting back into math more lately and that's one of the tougher things: I've forgotten a lot of what I can just take axiomatically and what I can't.
I've been getting back into math more lately and that's one of the tougher things: I've forgotten a lot of what I can just take axiomatically and what I can't.
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Re: Three externally tangent circles sharing a single point of tangency
Some attempts at proof might be tricky, because this conjecture is false in the degenerate case of a circle with radius 0. That is, you have two externally tangent circles, and a third 'circle' of radius 0 centered on the point of tangency. So the point of tangency is the entirety of the third circle.
It may also fail for the case of a 'circle' that passes through the point at infinity, (that is, treat the tangent line as a circle of infinite radius) depending on how you define internally vs externally tangent in that case.
But there are definitely no other exceptions.
It may also fail for the case of a 'circle' that passes through the point at infinity, (that is, treat the tangent line as a circle of infinite radius) depending on how you define internally vs externally tangent in that case.
But there are definitely no other exceptions.
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Re: Three externally tangent circles sharing a single point of tangency
arbiteroftruth wrote:It may also fail for the case of a 'circle' that passes through the point at infinity, (that is, treat the tangent line as a circle of infinite radius) depending on how you define internally vs externally tangent in that case.
But however you define it, it wouldn't be external to *both.*
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Re: Three externally tangent circles sharing a single point of tangency
Unless you define it in such a way that the circle centered at infinity has no interior.
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Re: Three externally tangent circles sharing a single point of tangency
Why, that's hardly a circle at all then.
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Re: Three externally tangent circles sharing a single point of tangency
GodShapedBullet wrote:I think this should help! If I can prove that the radius of a circle is perpendicular to a tangent line, I should be able to just add up the size of the angles and prove that they add up to more than 360 degrees. Let me mull on this.
You can prove this by showing that if the radius is not perpendicular to the tangent line, then you can make the radius shorter by moving the point on the circle slightly in one direction. More generally, by the extreme value theorem you can prove that the shortest line segment between a point and a differentiable curve (either infinitely long or forming a loop, so we don't have to worry about endpoints) is always perpendicular to the curve. For a circle every radius is a shortest line segment.
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Re: Three externally tangent circles sharing a single point of tangency
That's a really cool proof.
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Re: Three externally tangent circles sharing a single point of tangency
Now that this problem is solved: Would this also hold for spherical or hyperbolic geometries? Probably yes, but not sure.
Re: Three externally tangent circles sharing a single point of tangency
Not exactly answering the last question, but as long as you can say things like:
(1) If B is on a shortest path from A to C, and B is also on a shortest path from A to D, then either C is on the shortest path from A to D, or D is on the shortest path from A to C; and
(2) If the distance from A to B is x, and 0 < y < x, then there is a point C at distance y from A and at distance xy from B;
Then you can conclude:
(3) If three balls A, B, and C have pairwise intersections containing only a single point x, then at least one of A, B, or C consists only of x.
The argument would go that if two balls have any point of intersection, you can use (2) to show that they have a point of intersection on the shortest path between their centers, so that if they have only a single point of intersection, that point is on this shortest path. Then you can use (1) to argue that one of the shortest paths is a subset of the other shortest paths, so that the entire segment of the path between one of the centers and the common intersection point lies in one of the other circles.
(1) If B is on a shortest path from A to C, and B is also on a shortest path from A to D, then either C is on the shortest path from A to D, or D is on the shortest path from A to C; and
(2) If the distance from A to B is x, and 0 < y < x, then there is a point C at distance y from A and at distance xy from B;
Then you can conclude:
(3) If three balls A, B, and C have pairwise intersections containing only a single point x, then at least one of A, B, or C consists only of x.
The argument would go that if two balls have any point of intersection, you can use (2) to show that they have a point of intersection on the shortest path between their centers, so that if they have only a single point of intersection, that point is on this shortest path. Then you can use (1) to argue that one of the shortest paths is a subset of the other shortest paths, so that the entire segment of the path between one of the centers and the common intersection point lies in one of the other circles.
Re: Three externally tangent circles sharing a single point of tangency
In sphereical geometry, you will have trouble with "external"
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Re: Three externally tangent circles sharing a single point of tangency
Nicias wrote:In sphereical geometry, you will have trouble with "external"
Not really. You just need to give circles their centres, and then it should follow.
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Re: Three externally tangent circles sharing a single point of tangency
jestingrabbit wrote:Nicias wrote:In sphereical geometry, you will have trouble with "external"
Not really. You just need to give circles their centres, and then it should follow.
What about a great circle?
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Re: Three externally tangent circles sharing a single point of tangency
A great circle is no different than any other circle  a given circumference divides the sphere into two circles, with centers on opposite sides of the sphere. The only distinction is that a great circle doesn't have as many ways to define a "natural" circle to associate with the circumference. (Other circles can define the "natural" circle to be the smaller one, which matches people's intuitions most of the time.)
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Re: Three externally tangent circles sharing a single point of tangency
I can't really say that I see the small circle as any more natural than the large. If I want to describe where a missile can run out of fuel (a relatively natural question) whether its larger or smaller than a hemisphere is of little note.
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Re: Three externally tangent circles sharing a single point of tangency
I agree in general (both circles are valid), but if I draw a circle around myself on the ground and ask you where its center is, you're naturally going to point to my feet, not gesture vaguely to the opposite side of the world. The fact that some usecases have different notions of "natural" doesn't necessarily contradict the generic notion of which is more "natural"; the positive square root is the "natural" one, even tho there are usecases where you care about the negative one instead.
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