Can you provide a proof for the following proposition:

Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a common point of inscribed circle and side $AB$ , $Y$ a common point of inscribed circle and side $BC$ , $Z$ a common point of inscribed circle and side $CD$ , $W$ a common point of inscribed circle and side $DA$ and let $S$ be the intersection point of the line segments $XZ$ and $YW$ . I claim that the incenters of triangles $\triangle SAB$, $\triangle SBC$, $\triangle SCD$ and $\triangle SDA$ are concyclic.

GeoGebra applet that demonstrates this proposition can be found here.

My attempt:

Let $I_1,I_2,I_3,I_4$ be the incenters of triangles $\triangle SAB,\triangle SBC,\triangle SCD$ and $\triangle SDA$ , respectively. My idea is to apply Pythagorean theorem on triangles $\triangle SI_1I_2$, $\triangle SI_2I_3$,$\triangle SI_3I_4$ and $\triangle SI_4I_1$ in order to express lengths of the line segments $I_1I_2,I_2I_3,I_3I_4,I_4I_1$ using lengths of the line segments $SI_1,SI_2,SI_3,SI_4$ and then to apply Ptolomy's theorem on quadrilateral $I_1I_2I_3I_4$ , but I have difficulty proving that points $I_1,I_2,I_3$ and $I_4$ lie on the line segments $XZ$ and $YW$.

Can you provide a proof for the following proposition:

Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a common point of inscribed circle and side $AB$ , $Y$ a common point of inscribed circle and side $BC$ , $Z$ a common point of inscribed circle and side $CD$ , $W$ a common point of inscribed circle and side $DA$ and let $S$ be the intersection point of the line segments $XZ$ and $YW$ . I claim that the incenters of the $\triangle SAB$, $\triangle SBC$, $\triangle SCD$ and $\triangle SDA$ are concyclic.

GeoGebra applet that demonstrates this proposition can be found here.

My attempt:

Let $I_1,I_2,I_3,I_4$ be the incenters of   $\triangle SAB,\triangle SBC,\triangle SCD$ and $\triangle SDA$ , respectively. My idea is to apply Pythagorean theorem on   $\triangle SI_1I_2$, $\triangle SI_2I_3$,$\triangle SI_3I_4$ and $\triangle SI_4I_1$ in order to express lengths of the line segments $I_1I_2,I_2I_3,I_3I_4,I_4I_1$ using lengths of the line segments $SI_1,SI_2,SI_3,SI_4$ and then to apply Ptolomy's theorem on quadrilateral $I_1I_2I_3I_4$ , but I have difficulty proving that points $I_1,I_2,I_3$ and $I_4$ lie on the line segments $XZ$ and $YW$.

Can you provide a proof for the following proposition:

Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a common point of inscribed circle and side $AB$ , $Y$ a common point of inscribed circle and side $BC$ , $Z$ a common point of inscribed circle and side $CD$ , $W$ a common point of inscribed circle and side $DA$ and let $S$ be the intersection point of the line segments $XZ$ and $YW$ . I claim that the incenters of triangles $\triangle SAB$, $\triangle SBC$, $\triangle SCD$ and $\triangle SDA$ are concyclic.

GeoGebra applet that demonstrates this proposition can be found here.

My attempt:

Let $I_1,I_2,I_3,I_4$ be the incenters of triangles $\triangle SAB,\triangle SBC,\triangle SCD$ and $\triangle SDA$ , respectively. My idea is to apply Pythagorean theorem on triangles $\triangle SI_1I_2, \triangle SI_2I_3,\triangle SI_3I_4$ and $\triangle SI_4I_1$ in order to express lengths of the line segments $I_1I_2,I_2I_3,I_3I_4,I_4I_1$ using lengths of the line segments $SI_1,SI_2,SI_3,SI_4$ and then to apply Ptolomy's theorem on quadrilateral $I_1I_2I_3I_4$ , but I have difficulty proving that points $I_1,I_2,I_3$ and $I_4$ lie on the line segments $XZ$ and $YW$.

Can you provide a proof for the following proposition:

Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a common point of inscribed circle and side $AB$ , $Y$ a common point of inscribed circle and side $BC$ , $Z$ a common point of inscribed circle and side $CD$ , $W$ a common point of inscribed circle and side $DA$ and let $S$ be the intersection point of the line segments $XZ$ and $YW$ . I claim that the incenters of triangles $\triangle SAB$, $\triangle SBC$, $\triangle SCD$ and $\triangle SDA$ are concyclic.

GeoGebra applet that demonstrates this proposition can be found here.

My attempt:

Let $I_1,I_2,I_3,I_4$ be the incenters of triangles $\triangle SAB,\triangle SBC,\triangle SCD$ and $\triangle SDA$ , respectively. My idea is to apply Pythagorean theorem on triangles $\triangle SI_1I_2$, $\triangle SI_2I_3$,$\triangle SI_3I_4$ and $\triangle SI_4I_1$ in order to express lengths of the line segments $I_1I_2,I_2I_3,I_3I_4,I_4I_1$ using lengths of the line segments $SI_1,SI_2,SI_3,SI_4$ and then to apply Ptolomy's theorem on quadrilateral $I_1I_2I_3I_4$ , but I have difficulty proving that points $I_1,I_2,I_3$ and $I_4$ lie on the line segments $XZ$ and $YW$.