Let, $T$ be the nine-point circle of $\bigtriangleup ABC$. Let $l_y$ be the radical axis of $T,(Y)$ and $l_z$ be the radical axis of $T,(Z)$.And let $A_1$=$l_y \cap l_z$. And the nagel point and centroid of $\bigtriangleup DEF$ be $M$ and $P$ respectively.
$\text{Lemma 1:}$ $A_1FDE$ is a parallelogram.
$\text{Proof:}$ Simple angle and length chasing.
$\text{Lemma 2:}$ $H,P,M$ are collinear and $P$ divides $HM$ in the ration $1:2$.
$\text{Proof:}$ It is well known that $H$ is the incentre of $DEF$. And it is also well known that the centroid,incentre,nagel point of a triangle are collinear and the centroid divides line joining incentre and nagel point in the ratio $1:2$. So, the conclusion follows.
$\text{Lemma 3:}$ Let the midpoint of $EF$ be $U$. Then the reflection of $M$ under $U$ lies on the angle bisector of $\angle FDE$ i.e. $AD$.
$\text{Proof:}$ Let the reflection of $D$ under $H$ be $D'$. Applying Menelaus theorem to triangle $DPH$ and the points $D',M,U$ we get that they are collinear. Then applying Menelaus theorem to $DUD'$ and points $H,P,M,$ we get that $D'$ is the reflection of $M$ under $U$ and thus the conclusion follows.
$\text{Lemma 4:}$ $M$ lies on the angle bisector of $\angle FA_1E$.
$\text{Proof:}$ Since the reflection of $M$ under $U$ lies on the angle bisector of $\angle EDF$, applying symmetry to the whole figure gives us the result.
$\text{Lemma 5:}$ The angle bisector of $\angle FA_1E$ is perpendicular to YZ and BC.
$\text{Proof:}$ The angle bisector of $\angle FA_1E$ is parallel to $AD$ because of symmetry and the conclusion thus folllows as $AD$ is perpendicular to $BC$ and $YZ$.
$\text{Lemma 6:}$ $A_1$ lies on the radical axis of $(Y),(Z)$.
$\text{Proof:}$ Using radical axis theorem in circles $(Y),(Z),T$ the conclusion follows.
$\text{Lemma 7:}$ Radical axis of two circles is perpendicular to the line joining their centres.
$\text{Proof:}$ Well known.
Now, at last, by lemmas 4,5,6,7 we get that $A_1M$ is the radical axis of $(Y)$ and $(Z)$. And we come to a conclusion that $M$ is the radical centre of $(X),(Y),(Z)$.
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