Let the two quadrilaterals $ABCD$ and $EFGH$ been given:
Let's take these hypothesis:
$AD = EH$
$A\hat{B}D=A\hat{C}D=E\hat{F}H=E\hat{G}H$
$AC=EG$
The triangle $ABD$ is isosceles and equal to the triangle $EFH$
$A\hat{B}C + C\hat{D}A = E\hat{F}G+G\hat{H}E=180°$
$B\hat{C}D+ D\hat{A}B=F\hat{G}H+H\hat{E}F=180°$
Thesis: $CD = GH$.
Even if there are all these hypotheses I'm finding difficulties. It's simple to prove that $B\hat{A}C= B\hat{D}C$ and $F\hat{E}G= F\hat{H}G$, but for some reason I can't prove that $B\hat{A}C=F\hat{E}G$, even if I know it's true.
Can you give me a hand? Thank you!
$ABD=EFH$, since they both are isosceles, with the same base and the same opposite angle. $C$ belongs to the circumcircle of $ABD$ as well as $G$ belongs to the circumcircle of $EFH$, and since $EG=AC$, the two quadrilaterals overlap.