Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle with semiperimeter
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
and inradius
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
. The semicircles with diameters
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
are drawn on the outside of the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
. The circle tangent to all of these three semicircles has radius
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
. Prove that
Alternative formulation. In a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, construct circles with diameters
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
, and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, respectively. Construct a circle
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
externally tangent to these three circles. Let the radius of this circle
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
be
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
.
Prove:
![\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r](/media/m/3/3/e/33ea79cb660b9ba799a6d7d41b84798c.png)
, where
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
is the inradius and
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
is the semiperimeter of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
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Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC$, $CA$, $AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all of these three semicircles has radius $t$. Prove that
$$\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r.$$
Alternative formulation. In a triangle $ABC$, construct circles with diameters $BC$, $CA$, and $AB$, respectively. Construct a circle $w$ externally tangent to these three circles. Let the radius of this circle $w$ be $t$.
Prove: $\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r$, where $r$ is the inradius and $s$ is the semiperimeter of triangle $ABC$.