Dear Peter and All My Friends,

Thank you for your interest and very good study for my problem. When I posted it, I thought I can find one nice pure geometrical proof and construction for it. But until now it makes me headache.

After reading your discussion, I think it can not be geometrically constructed generally. In any case, we can hope that with some particular arcs, it is can be constructed and may be related golden ratio (It is very happy case!)

Please note that there are two cases: when right angle vertex on chord and when right angle vertex on arc. May be there is one simple geometrical proof that the right vertex of maximal area triangle must be on arc (as in Peter discussion).

Best regards,
Bui Quang Tuan