Problem 1: Let $ABCD$ be a quadrilateral, $E$ be the midpoint of $BC$, $F$ be the midpoint of $AD$. The line though $E$ and perpendicular to $BC$ meets the line through $F$ and perpendicular $AD$ at $G$. Show that:
1-$FG$, and the line through $B$ perpendicular to $AG$, and the line through $C$ perpendicular to $DG$ are concurrent at $P_{bc}$.
2-If $ABCD$ be a cyclic quadrilateral, show that $P_{ab}P_{bc}P_{cd}P_{da}$ be a tangential quadrilateral
Problem 2: Let $ABCD$ be a tangential quadrilateral, let $A_1,B_1,C_1,D_1$ be midpoint of $AB,BC,CD,DA$ respectively. Let $A_2,B_2,C_2,D_2$ lie on $AB,BC,CD,DA$ respectively. Such that $A_1C_2, B_1D_2,C_1A_2,D_1B_2$ perpendicular to $CD,DA,AB,BC$ respectively. Then the quadrilateral bounded by $A_1C_2, B_1D_2,C_1A_2,D_1B_2$ is a tangential quadrilateral.
1-$FG$, and the line through $B$ perpendicular to $AG$, and the line through $C$ perpendicular to $DG$ are concurrent at $P_{bc}$.
2-If $ABCD$ be a cyclic quadrilateral, show that $P_{ab}P_{bc}P_{cd}P_{da}$ be a tangential quadrilateral
Problem 2: Let $ABCD$ be a tangential quadrilateral, let $A_1,B_1,C_1,D_1$ be midpoint of $AB,BC,CD,DA$ respectively. Let $A_2,B_2,C_2,D_2$ lie on $AB,BC,CD,DA$ respectively. Such that $A_1C_2, B_1D_2,C_1A_2,D_1B_2$ perpendicular to $CD,DA,AB,BC$ respectively. Then the quadrilateral bounded by $A_1C_2, B_1D_2,C_1A_2,D_1B_2$ is a tangential quadrilateral.
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