<Cir-202-797>, hint

Line QB cut AP at R, O be the center of the circle.

Let <ORA=<OBA=x, Since triangles OPA and OQB are congruent, 

<AOB=180-2x=<POQ, OP=OQ, <OPQ=<OQP=x, and <OBA=x. 

Hence O,Q,S,B are concyclic and <OBQ=<OSQ=90.

Finally PS=QS (OPQ is isosceles, )

<Cir-2026-797>, tangent midpoint 

<Singapore MO 1998>

<Tri-2026-2303>, hint

[ABC]; area of triangle ABC 

[AMN]+[BNL]+[CLM]+[LMN]=[ABC]

sin60{ AMxAN+BLxBN+CLxCM}+[LMN]=sin60 BC xAB

AMxAN+BLxBN+CLxCM<BCxAB=BCxBC

<Tri-2026-2303> equilateral, area

<Singapore MO 1997>

 

<Qua-2026-811>, hint