I think Yuri wants to prove that the teacher had a relationship with the husband so he can ruin his sister's marriage, but I think it's obvious they were just friends so really Yuri you're not just an assh0le but also wasting your time.
He’s in love with his sister so his goal atm is to prove this guy isn’t who he says he is, even though the guys seems to just be nice. Yuri is young and assuming no one will be as good as his sister.
I think Yuri wants to prove that the teacher had a relationship with the husband so he can ruin his sister's marriage, but I think it's obvious they were just friends so really Yuri you're not just an assh0le but also wasting your time.