When I read the new paper 100 CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS-section 7, I remember that I posed some problem associated with tangential quadrilateral from 2014 in here and 2015 in here but have no solution until now. This problem is alike to Christopher Bradley’s conjecture.
I am looking for a proof of the problem.
Let $ABCD$ be a tangential quadrilateral with an inscribed circle $(I)$, Let $P$ be the intersection of the perpendicular bisector of $AC$ and the perpendicular bisector of $BD$. Let $I_A$, $I_B$, $I_C$, $I_D $ are four incenter to triangles $\triangle PAB$, $\triangle PBC$, $\triangle PCD$, $\triangle PDA$ respectively. Let $E_A$, $E_B$, $E_C$, $E_D$ are four excenters to triangles $\triangle PAB$, $\triangle PBC$, $\triangle PCD$, $\triangle PDA$ opposite the vertex $P$.
The incenters $I_A, I_B, I_C, I_D$ are concyclic.
The excenters $E_A, E_B, E_C, E_D $ are concyclic.
The centers $I_A, I_C, E_A, E_C$ are concyclic.
The centers $I_B, I_D, E_B, E_D$ are concyclic.
Center of four circles: $(I_A, I_B, I_C, I_D)$, $(E_A, E_B, E_C, E_D)$, $(I_A, I_C, E_A, E_C)$, $(I_B, I_D, E_B, E_D)$ are an orthocentric system.
$P$ lie on the line joining centers of two circles $(I_A, I_B, I_C, I_D)$ and $(E_A, E_B, E_C, E_D)$.
Note that: the problem by Telv Cohl from 2015 here is exactly the same problem as mine from 2014 here.
