Let's take for example $\triangle ABC$ with $\angle A = \angle B = 1^o$. How can a triangle like this have a circumcircle? My confusion is with triangles like this in general, with very long sides.
How can every triangle have a circumcircle
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$\begingroup$
geometry
triangles
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4How many non-collinear points do you need so that a unique circle passes through them? :) – 2012-12-18
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1No matter how long you make the sides, I can make a circle big enough to circumscribe your triangle. We have all of infinity to play with, so size matters not (to paraphrase Yoda). – 2012-12-18
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0Maybe visualise it like this: imagine any triangle with a circumcircle, with one vertex at the top. Now imagine moving the other two points round the circle - getting nearer to the top. That way you can get a triangle with 2 angles of just one degree. Then just enlarge the diagram. – 2012-12-18
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1Oh, I think I visualize it now.. so those points would be relative close together on the circle? – 2012-12-18
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0Yes, they would need to be comparatively close together. – 2012-12-18
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0Note that the smallest circle containing an obtuse angled triangle will only go through two of the vertices and will be smaller than the circumcircle; in your example, much smaller. But there is still a circumcircle – 2012-12-18
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0Would this also mean that the points on circumcircle of an equilateral triangle are furthest apart? – 2012-12-18
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0Once you understand this, and if you know a little trigonometry, try this problem. If a triangle has sides 1, 1, and a little less than 2, so that its angles are $1^\circ$, $1^\circ$, and $88^\circ$, then what is the radius of the circumcircle? – 2012-12-18
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0@GEdgar $1+1+88-90 ;)$. – 2012-12-18
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0OK, we need 1,1,178. – 2012-12-18