Let be a triangle. How to find a bound of ratio of circumradius and inradius. One may think that the edge case is in a triangle that is somehow special. It’s trivial, that when one is dealing with equlateral case, since its incircle and circumcircle are cocentric. To find the ratio, a handful observation comes to mind - the common center of those circles is the centroid of and since centroid divides every median into two pieces with one of them the double of the other, the ratio is exactly .
This gives an impression, that is an extremal value. One can easily check that if it actually is an extreme, it is the minimal value. Quick look on a isosceles right triangle assures this is a valid constraint.
What happens in a general case? Of course, there are a lot of proofs, some of them applying to -dimensional space, however I’d like to present a method that uses algorithmic, approximation techniques.
Let be a constant circle. Every triangle can be generated from three points of this circle, with a help of homothety. Therefore, if we find the triangle with maximal inradius, and it turns out to be equilateral, .
Take a chord of as constant. Point varies on the circle. Angle is therefore constant. Since is the incenter, it lies on the angle bisectors of and . From the relation follows that is constant. Therefore, is constant too. is a triangle with to of angles equal to and respectively. This proves that is constant too. This means, that with arbitrary , lie on a circle that is not dependent on . The inradius of is the distance from to and since , , share the same circle, inradius is greatest when , i.e. . In other words, is isosceles with as base. If we can prove that applying this method of leaving one of angles intact and making the other two equal cyclically to triangles , , , , makes the triangle more and more equilateral, it would mean that none of triangles has greater ratio. In other words, triples converge to and inradius is strictly increasing.
I will facilitate my thought with the AM-GM inequality which states that for any sequence of non-negative numbers , the relation holds:
What I really need is the proof for or . That’s why I won’t provide full proof for all cases. My version is a slightly modified proof by Augustin-Louis Cauchy. If , proof is obvious. Let’s prove it works when then.
Squareing both sides, we get
This holds for any real numbers .
Now, let’s see how it works, when . For proof is complete. Assuming thesis is correct for , I’ll show why it works for .
Applying it again, for and , we get
First, I need to show a little lemma:
Multiplying both sides by finishes the lemma’s proof:
Now, let’s try to do some mangling with symbols:
We can exponentiate it sidewise:
Divide it by the content of the parenthesis to get
i.e. AM-GM for .
Coming back to proof of original problem - I’ve already shown that substituting and by their arithmetic mean increases the inradius. From AM-GM follows, that
If the growth of geometric mean of is associated with increase of inradius, this will be the end of the proof, because the mean is bounded from the ceiling by a constant ( in this case). So, equivalent inequality follows:
Dividing both sides by , we get
or, in other words
Square root it:
which holds from AM-GM. Therefore, as angles of are nearing to degrees, inradius strictly increases. Since when they are equal to each other, the ratio is equal to , and every other case has this value smaller, it is the greatest bound, that was to find.