5) Let ABCD be a convex quadrilateral.Let diagonals AC and BD intersect at P. Let PE,PF,PG and PH are altitudes from P on the side AB,BC,CD and DA respectively. Show that ABCD has a incircle if and only if \( \frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH} \).

1) Let ABC be a right-angled triangle with \( \angle{B}=90^{\circ} \). Let BD is the altitude from B on AC. Let P,Q and be the incenters of triangles ABD,CBD and ABC respectively. Show that circumcenter of triangle PIQ lie on the hypotenuse AC.

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