1. Let 
$ABCD$ be a cyclic quadrilateral. Show that the orthocenters of the triangles $ABC, BCD. CDA, DAB$ 
are vertices of a quadrilateral congruent to $ABCD$ and show that the centroids of the same triangles are the vertices of a cyclic quadrilateral.
$ABCD$ be a cyclic quadrilateral. Show that the orthocenters of the triangles $ABC, BCD. CDA, DAB$ are vertices of a quadrilateral congruent to $ABCD$ and show that the centroids of the same triangles are the vertices of a cyclic quadrilateral. 
2. $ABCD$ is a cyclic quadrilateral with $BC=CD$
. The diagonals $AC$ 
and $BD$ 
intersect at 
$E$. Let 
$X, Y, Z, W$ be the incenters $ABE, ADE, ABC, ADC$ of triangles 
respectively. Show that $X, Y, Z, W$ are concyclic iff 
$AB = AD$.
$ABCD$ is a cyclic quadrilateral with $BC=CD$. The diagonals $AC$ and $BD$ intersect at $E$. Let $X, Y, Z, W$ be the incenters $ABE, ADE, ABC, ADC$ of triangles respectively. Show that $X, Y, Z, W$ are concyclic iff $AB = AD$. 3. $A$ 
and 
$B$ are fixed points on a plane and $L$ 
is a line passing through $A$ and not $B$. $C$ 
is a variable point moving from $A$ toward infinity along a half-line of $L$. The incircle of 
$\triangle{ABC}$ touches 
$BC$ at 
$D$ and $AC$ at $E$. Show that the line 
$DE$ passes through a fixed point.
and $B$ are fixed points on a plane and $L$ is a line passing through $A$ and not $B$. $C$ is a variable point moving from $A$ toward infinity along a half-line of $L$. The incircle of $\triangle{ABC}$ touches $BC$ at $D$ and $AC$ at $E$. Show that the line $DE$ passes through a fixed point. 4. Let $\triangle{ABC}$ be given with circumradius $R$
. Let 
$K_1$ and 
$K_2$ be the circles which pass through $C$ and are tangent to $AB$ at 
$A$ and $B$ respectively. Let 
$K$ be the circle of radius 
$r$ which is tangent to $K_1, $K_2$ 
and $AB$.
be given with circumradius $R$. Let $K_1$ and $K_2$ be the circles which pass through $C$ and are tangent to $AB$ at $A$ and $B$ respectively. Let $K$ be the circle of radius $r$ which is tangent to $K_1, $K_2$ and $AB$. (a) Express $r$ in terms of 
$a, b, c$ and $R$.
$r$ in terms of $a, b, c$ and $R$. 
$r= \frac{1}{4}R$, find 
$\angle{C}$.
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