<Qua-2026-813>, hint

Let <BAD=2x, then <BAK=<BKA=<CKL=<CLK=x 

Let O be the circumcenter of triangle CKL, OK=OC (radius), <KOC=2KLC=2x.

BK=BA=CD, <OCD=<BCD+<OCK=2x+90-x=90+x, <OKB=180-<OKC(90-x)=90+x

Hence triangles OCD and OKB are congruent and <OBC=<ODC and O,B,C,D are cyclic.

<Qua-2026-813>, parallelogram, cyclic

<South Africa  MO 1999>

<Tri-2026-2320>, hint

Let S be the point of concurrence  of  chords AD,BE,CF and O the center of circle.

Since OP, OQ OR are perpendicular to AD,BE,CF respectively O, P, Q,R are concyclic

Let <CSD=x, <CSF=y, <ASF=z, then by angle chasing <PRQ=x, <PQR=z, <QPR=y

<GHD=x, <GDH=y, <DGH=z

<Tri-2026-2320>, general, similar

<South Africa MO 1998>

 

<Tri-2026-2319>, hint