I would need this for a proof.

enter image description here

I have a two circles intersecting at two points P and Q. We draw a segment between this two points and we draw a segment $AB$ which is perpendicular to $PQ$ and goes through P. Then we draw another segment $UV$ which goes through P and through a point $U$ in the first circle and $V$ in the second.

Can I say something about the following triangles that I drew in my image? Especially about their similarity?

I would only please need a hint, I want to solve this myself.

I thank everyone for help and am sorry for the not so good picture.

EDIT:I am looking for some similarity of UOP,AOQ,QHW and PHB. If there is any other way I can see them to find some similarity I would be thankful for a hint.

  • $\begingroup$ Which triangles are you interested in? Please edit the question to tell us - don't clarify in comments. One suggestion: does the similarity look plausible to you if one of the circles is really large? $\endgroup$ Commented May 1, 2021 at 19:50

1 Answer 1


Hint: Angles that subtend the same arc are congruent so $\angle U \cong \angle A$ and $\angle W \cong \angle B$

  • $\begingroup$ Thank you so much. Very helpful. I will try to take it from here. $\endgroup$ Commented May 1, 2021 at 19:56
  • $\begingroup$ would the arcs in the intersection also be the same? $\endgroup$ Commented May 1, 2021 at 20:05
  • $\begingroup$ @vintagestyle: if you are talking about $PQ$ no, they would not be the same as circles have different radii $\endgroup$
    – Vasili
    Commented May 1, 2021 at 22:00
  • $\begingroup$ I would need to prove something regarding expressing $UW$ depending on the cosine of the angle $HPB$. Would you have any advice, especially to which triangles I should examine. Or should I maybe ask this as a new question? $\endgroup$ Commented May 4, 2021 at 11:15
  • $\begingroup$ @vintagestyle: what exactly do you need to prove? $\endgroup$
    – Vasili
    Commented May 4, 2021 at 13:47

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