View topic - Some properties of the circumscribed quadrilateral • Art of Problem Solving

Some properties of the circumscribed quadrilateral.
Let $ABCD$ be a quadrilateral which is circumscribed about the circle $w=C(I,r)$ .
Denote: $a=AB$ , $b=BC$ , $c=CD$ , $d=DA$ , $e=AC$ , $f=BD$ ;
the middlepoints $M$ , $N$ of the diagonals $AC$ , $BD$ respectively;
$E\in AC\cap BD$ , $F\in AB\cap CD$ , $G\in AD\cap BC$ ;
the tangent - points $X\in AB$ , $Y\in BC$ , $Z\in CD$ , $T\in DA$ of the incircle $w$ ;
$U\in XT\cap YZ$ , $V\in XY\cap ZT$ ; the area $S=[ABCD]$ , $2p=a+b+c+d$ . Then:

$1.$ The quadrilateral $ABCD$ is circumscribed $\Longleftrightarrow a+c=b+d$ (Pithot).
$2.\ E\in XZ\cap YT$ (Newton); $I\in MN$ (Newton's line).
$3.\ U\in BD$ and $(D,E,B,U)$ is a harmonic division.
$V\in AC$ and $(A,E,C,V)$ is a harmonic division.
$4.$ The quadrilateral $IEUV$ is orthocentric, i.e. $UI\perp EV$ and $VI\perp EU$ .

$5.$ If the quadrilateral $ABCD$ is circumscribed and inscribed in the circle $e=C(O,R)$ , then:
$5.1.\ ef=2r\left(r+\sqrt{4R^{2}+r^{2}}\right)$ , $OI^{2}=R^{2}+r^{2}-r\sqrt{4R^{2}+r^{2}}$ (Durrande).
$5.2\ E\in XZ\cap YT$ (Newton); $\frac{IM}{IN}=\frac{e}{f}$ ; $\frac{1}{e}+\frac{1}{f}=\frac{p}{4Rr}$ .
$5.3.$ The power of the point $E$ w.r.t the circle $e$ is $p_{e}(E)=\left(\frac{ef}{4R}\right)^{2}$ ;
$5.4.$ The power of the incenter $I$ w.r.t. the circumcircle $e$ is $p_{e}(I)=\frac{8R^{2}r^{2}}{ef}$ .

Remark. Maybe now the following problem seems more easily:
See www.artofproblemsolving.com/Forum/viewtopic.php?t=69101

Some properties of the circumscribed quadrilateral

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