I found this problem in a section of an old notebook where i used to write down weird problems i came across and that I didn't know how to solve. Long story short i rediscovered this notebook a week ago and managed to solve most geometry problems, with the exception of this .

$n\ge 4$ $A_1 , A_2 ,... , A_n$ concyclic; $h$ is the set of the orthocenters of the triangles determined by these points, which are noted (the orthocenters) $ H_1 , H_2 , ...$ .
Show that : $$\sum_{H_a,H_b∈h} H_aH_b ≥\frac{(n-2)(n-3)}{2} \sum_{1≤i,j≤n} A_iA_j $$ and calculate when does the equality take place.

Note  : $ H_aH_b and A_iA_j $ are the length of the segments

I am completely stumped by this problem.

I found this problem in a section of an old notebook, where I used to write down weird problems I came across and that I didn't know how to solve. Long story short, I rediscovered this notebook a week ago and managed to solve most geometry problems, with the exception of the following.

Let $n\ge 4$, and let $A_1, A_2,\dotsc, A_n$ be concyclic. Let $h$ be the set of the orthocenters of the triangles determined by these points, and let us label its elements (the orthocenters) as $H_1, H_2, \dotsc$

Show that: $$\sum_{H_a,H_b∈h} H_aH_b ≥\frac{(n-2)(n-3)}{2} \sum_{1\leq i,j\leq n} A_iA_j,$$ and determine when equality takes place.

Note: $H_aH_b$ and $A_iA_j$ are the length of the segments.

I am completely stumped by this problem.

I found this problem in a section of an old notebook where i used to write down weird problems i came across and that I didn't know how to solve. Long story short i rediscovered this notebook a week ago and managed to solve most geometry problems, with the exception of this .

$n\ge 4$ $A_1 , A_2 ,... , A_n$ concyclic; $h$ is the set of the orthocenters of the triangles determined by these points, which are noted (the orthocenters) $ H_1 , H_2 , ...$ .
Show that : $$\sum_{H_1,H_2∈h} H_1H_2 ≥\frac{(n-2)(n-3)}{2} \sum_{1≤i,j≤n} A_iA_j $$ and calculate when does the equality take place.

Note : $ H_1H_2 and A_iA_j $ are the length of the segments

I am completely stumped by this problem.

I found this problem in a section of an old notebook where i used to write down weird problems i came across and that I didn't know how to solve. Long story short i rediscovered this notebook a week ago and managed to solve most geometry problems, with the exception of this .

$n\ge 4$ $A_1 , A_2 ,... , A_n$ concyclic; $h$ is the set of the orthocenters of the triangles determined by these points, which are noted (the orthocenters) $ H_1 , H_2 , ...$ .
Show that : $$\sum_{H_a,H_b∈h} H_aH_b ≥\frac{(n-2)(n-3)}{2} \sum_{1≤i,j≤n} A_iA_j $$ and calculate when does the equality take place.

Note : $ H_aH_b and A_iA_j $ are the length of the segments

I am completely stumped by this problem.