AoPS Academy
be an acute triangle with incenter 
and circumcenter 
. The incircle touches sides 
and 
at 
and 
respectively, and 
is the reflection of 
over . The circumcircles of 
and 
meet at 
, and the circumcircles of 
and meet at a point 
, where 
is the midpoint of 
. Prove that if 
and meet at 
, then 
.
and 
Prove that
such that divides 
but 
does not divide 
.
be a tetrahedron having each sum of opposite sides equal to 
. Prove that![\[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\]](http://latex.artofproblemsolving.com/8/a/6/8a6ff45706ced02e639c940624c2bb911fbcf852.png)
are the inradii of the faces, equality holding only if is regular.
denote 
, 
, 
, 
, 
and 
. Also let 
, 
, 
and 
be the inradii of the triangles (faces of the tetrahedron) 
, 
, 
and respectively . For each of these triangles apply the known inequality of Mitrinovic i.e. 
, where 
and 
are the semiperimeter and the inradius of a given triangle . Indeed, ![$\begin{array}{ccc} 3\sqrt 3\cdot\left(r_A+r_B+r_C+r_D\right)\ \le\ \frac 12\cdot\left[\left(b+c+f\right)+\left(c+d+e\right)+\left(d+a+f\right)+\left(a+b+e\right)\right]=a+b+c+d+e+f\end{array}$](http://latex.artofproblemsolving.com/b/f/a/bfa8129fa2ade4017d1ff8561aba311a31966a24.png)
and since 
, the last inequality finally reduces to : 
. Equality holds iff the triangles , , and are equilateral i.e when is a regular tetrahedron .
![\[[ABc]=rs=\sqrt{s(s-a)(s-b)(s-c)}.\]](http://latex.artofproblemsolving.com/3/e/c/3ece5d427c240610f45703552a198ada8f9d9565.png)
By AM-GM, we have ![\[\sqrt[3]{(s-a)(s-b)(s-c)}\le\frac{s-a+s-b+s-c}{3}=\frac{s}{3}.\]](http://latex.artofproblemsolving.com/c/0/c/c0cfd41d54ef1fc34dd57d6efb486bb1fbb9924c.png)
Therefore ![\[rs=s\sqrt{(s-a)(s-b)(s-c)}{s}=\sqrt{3}\left(\sqrt[3]{(s-a)(s-b)(s-c)}\right)^{\frac{3}{2}}\le\sqrt{s}\sqrt{\frac{s^3}{27}},\]](http://latex.artofproblemsolving.com/6/8/c/68ccfff94058aa82ad6e17cbe3ecc4b0e54d739b.png)
which implies that for any triangle we have ![\[r\le\frac{s\sqrt{3}}{3}.\]](http://latex.artofproblemsolving.com/2/1/b/21be011c291b84681c94fc258d042845e3a54462.png)
Now applying this to each face of the tetrahedron yields ![\[\sum r_A\le\frac{1}{3\sqrt{3}}\sum s_A\]](http://latex.artofproblemsolving.com/3/2/3/323aeca7898c6b06cfcb3783f2b7ba60122cb8e1.png)
where 
denote the semiperimeters of the faces. Since each side is by exactly 
triangles, we have 
. Hence ![\[r_A+r_B+r_C+r_D\le\frac{\sqrt{3}}{3}\]](http://latex.artofproblemsolving.com/8/5/e/85e43a3f467bf8475b4ff0cf5327f0d0dd6316a8.png)
which equality when 
. This follows from AM-GM, where 
and the triangle is equilateral, so the tetrahedron is regular.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
