I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:

Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. 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 $B_{i+1}=B_1$. Then show that:

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

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

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.