证明：$\because$圆周角$\angle A$和$\angle C$都对着$\widehat {BD}$，
$\therefore \angle A=\angle C$，
在$\triangle ADP$和$\triangle CBP$中，
$\left\{\begin{array}{l}{∠A=∠C}\\{AP=CP}\\{∠APD=∠CPB}\end{array}\right.$，
$\therefore \triangle ADP$≌$\triangle CBP\left(ASA\right)$，
$\therefore BP=DP$，
$\because AP=CP$，
$\therefore AP+BP=CP+DP$，
即$AB=CD$.

证明：$\because$圆周角$\angle A$和$\angle C$都对着$\widehat {BD}$，
$\therefore \angle A=\angle C$，
在$\triangle ADP$和$\triangle CBP$中，
$\left\{\begin{array}{l}{∠A=∠C}\\{AP=CP}\\{∠APD=∠CPB}\end{array}\right.$，
$\therefore \triangle ADP$≌$\triangle CBP\left(ASA\right)$，
$\therefore BP=DP$，
$\because AP=CP$，
$\therefore AP+BP=CP+DP$，
即$AB=CD$.