Thank you for your explanation of the emphases; it's simpler than I expected it to be.
It seems to me that Sujeet's and Peter's solutions are distinct and also that I have two distinct non-solutions which meet the angles given but not the no-isosceles requirement. That's four! I am looking forward to the all-in-one solution you will demonstrate.
Also, I'm hoping that someone will take up Sujeet's desire for problems using angles which are multiples of pi/7. Are these adventitious angles? I have much to learn.