I proposed my conjecture, [it is strengthened version of the Erdős–Mordell inequality](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Mordell_inequality) as following:

Let $A_1A_2.....A_n$ be a [cyclic Polygon](http://mathworld.wolfram.com/CyclicPolygon.html) and $B_1B_2....B_n$ be the [its tangential polygon](http://mathworld.wolfram.com/TangentialPolygon.html). Let P be an arbitrary point inside of $A_1A_2.....A_n$. Let $D_i$ be the distances from $P$ to $A_iA_{i+1}$, for $i=1,...,n$ and $A_{i+1}=A_1$, and $d_i$ be the distances from $P$ to $B_iB_{i+1}$, for $i=1,...,n$ and $B_{i+1}=A_1$. Then:


$$\sum_{1}^{n}{ D_i} \ge sec{\frac{\pi}{n}}\sum_{1}^{n}{d_i} $$

Equality holds when $A_1A_2....A_n$ be the regular polygon.

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/RSUPN.png