<Tri-2026-2321>, hint

 

<Tri-2026-2321>, general, incenter

<South Africa MO 2002>

 

<Qua-2026-817>, hint

 

<Qua-2026-817>,square, angle

<South Africa MO 2002>

 

<Qua-2026-816>, hint

AC and BD intersect at O. Let OA=a, OB=b, OC=c, OD=d, <AOB=X

By cosine 2nd law   and given condition you can get

(ab+bc+cd+da) cos X=0, cosX=0, X=90, ACㅗBD