Let $\triangle ABC$ be a scalene triangle with point $D$ inside it. Let $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the incenters of the triangles $\triangle BDO$ and $\triangle CDN$ be $D1$. Similarly define points $D2$ and $D3$.
It is true: $AN + BO + CM = AO + CN + BM$.

Prove that $D, D1, D2, D3$ are on the same circle using Ptolemy's trigonometric theorem.

I tried to use Ptolemy for $D, D1, D2, D3$ points but the angles were not right. I computed the distance between $D$ and incenters. I tried to compute the angles between $D1$ and $D$, and all the others but they do not seem to get simplified.
I used Ceva for $D$.
I tried to prove that $D$ is the incenter of $\triangle ABC$.
But anything that I used was too complicated to compute.

Can you find examples of Point $D$?

Here is a picture. Can $D1$ and $D$ be colinear? I could not prove it, but in Geogebra it seems so.

Let $\triangle ABC$ be a scalene triangle with point $D$ inside it. Let $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the incenters of the triangles $\triangle BDO$ and $\triangle CDN$ be $D1$. Similarly define points $D2$ and $D3$.
It is true: $AN + BO + CM = AO + CN + BM$.

Prove that $D, D1, D2, D3$ are on the same circle using Ptolemy's trigonometric theorem.

I tried to use Ptolemy for $D, D1, D2, D3$ points but the angles were not right. I computed the distance between $D$ and incenters. I tried to compute the angles between $D1$ and $D$, and all the others but they do not seem to get simplified.
I used Ceva for $D$.
I tried to prove that $D$ is the incenter of $\triangle ABC$.
But anything that I used was too complicated to compute.

Can you find examples of Point $D$?

Here is a picture. Can $I1$, $I2$ and $D$ be colinear? I could not prove it, but in Geogebra it seems so.

Let $\triangle ABC$ be a scalene triangle and point $D$ inside it. $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the incenters of the triangles $\triangle BDO$ and $\triangle CDN$ be $D1$. Similarly define points $D2$ and $D3$. It is true: AN + BO + CM = AO + CN + BM.

Prove that D, D1, D2 and D3 are on same circle using Ptolemy's trigonometric theorem.

I tried to use Ptolemy for D, D1, D2, and D3 points but the angles where not ok. I computed the distance between D and incenters. I tried to compute the angles between D1 and D, and all the others but they seems not get simplified. I used Ceva for D. I tried to prove that D is the incenter of $\triangle ABC$. But anything that I used it was too complicated to compute.

Can you find examples of D point?

Here is a picture. Can D1 and D be colinear? I couldn't prove it, but in Geogebra seems so.

Let $\triangle ABC$ be a scalene triangle with point $D$ inside it. Let $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the incenters of the triangles $\triangle BDO$ and $\triangle CDN$ be $D1$. Similarly define points $D2$ and $D3$. 
It is true: $AN + BO + CM = AO + CN + BM$.

Prove that $D, D1, D2, D3$ are on the same circle using Ptolemy's trigonometric theorem.

I tried to use Ptolemy for $D, D1, D2, D3$ points but the angles were not right. I computed the distance between $D$ and incenters. I tried to compute the angles between $D1$ and $D$, and all the others but they do not seem to get simplified. 
I used Ceva for $D$. 
I tried to prove that $D$ is the incenter of $\triangle ABC$. 
But anything that I used was too complicated to compute.

Can you find examples of Point $D$?

Here is a picture. Can $D1$ and $D$ be colinear? I could not prove it, but in Geogebra it seems so.

Let $\triangle ABC$ be a scalene triangle and point $D$ inside it. $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the centers of the inscribed circles in the triangles $\triangle BDO$ and $\triangle CDN$ be $D1$. Similarly define points $D2$ and $D3$. It is true: AN + BO + CM = AO + CN + BM.

Prove that D, D1, D2 and D3 are on same circle using Ptolemy's trigonometric theorem.

I tried to use Ptolemy for D, D1, D2, and D3 points but the angles where not ok. I computed the distance between D and incenters. I tried to compute the angles between D1 and D, and all the others but they seems not get simplified. I used Ceva for D. I tried to prove that D is the incenter of $\triangle ABC$. But anything that I used it was too complicated to compute.

Can you find examples of D point?

Here is a picture. Can D1 and D be colinear? I couldn't prove it, but in Geogebra seems so.

Let $\triangle ABC$ be a scalene triangle and point $D$ inside it. $AD$, $BD$ and $CD$ intersect $BC$, $CA$ and $AB$ in $M$, $N$ and $O$. Let the midpoint of the segment that connects the incenters of the triangles $\triangle BDO$ and $\triangle CDN$ be $D1$. Similarly define points $D2$ and $D3$. It is true: AN + BO + CM = AO + CN + BM.

Prove that D, D1, D2 and D3 are on same circle using Ptolemy's trigonometric theorem.

I tried to use Ptolemy for D, D1, D2, and D3 points but the angles where not ok. I computed the distance between D and incenters. I tried to compute the angles between D1 and D, and all the others but they seems not get simplified. I used Ceva for D. I tried to prove that D is the incenter of $\triangle ABC$. But anything that I used it was too complicated to compute.

Can you find examples of D point?

Here is a picture. Can D1 and D be colinear? I couldn't prove it, but in Geogebra seems so.