Inscribed, circumscriebed circles problem

[drowsy kelp]

Everyone knows we can easily find these 2 if we are given a triangle, but what about the other way around? Given 2 circles, the inscribed and circumscribed circle, can we find back the original triangle?

[lime shuttle]

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[rotund willow]

currently operating on 1 brain cell, but yeah I'd think so

the inscribed circle is has 3 tangents, and will touch each of those so that the angular bisectors all intersect the centre of the circle

because the origin of the inscribed circle is the incenter of the triangle

and then you get your vertexes from the circumcircle

[rotund willow]

this is because if you take the perpendicular bisector of all 3 sides, they overlap at the 'centre of mass'

@drowsy kelp

obviously you'd have to have the inscribed and circumcircles in the correct place relative to eachother

[drowsy kelp]

Then

Can any 2 circles form a triangle, given that one is the "inscribed" and the other one the "circumscribed" one?

[rotund willow]

there will be some constraints on that though

but icba to formalise that

the issue is that if you have two circles with very close radii then you will end up with angles between lines summing to greater than 180°

which obviously can't be a triangle anymore

so any 2 circles? no

[drowsy kelp]

Hmm, would the position in space not affect if a triangle is poasible to form, or just the radii?

[rotund willow]

the position would effect it a bit

easiest constraint is the inscribed circle must be entirely contained within the circle

[drowsy kelp]

True

[rotund willow]

but even a not similar radius could make it impossible, you see the problem here

if they're concentric then there will always be either no possible triangle or an equilateral

but that's basic

[rotund willow]

this is specifying down a theorem (?) that any two circles where one is entirely contained within another has a polygon to which they are the inscribed and circumscribed circles

[drowsy kelp]

CraZy

Wish you could remember its name

[drowsy kelp]

+close