Two circles cut each other at the points A and B. A straight line is drawn through A, meeting the first circle again at P and the second circle again at Q. The line is now rotated about A, so P and Q move. Which quantity stays the same throughout?

In the first circle ∠APB is an inscribed angle standing on the fixed chord AB, so it keeps the same value as P moves along its arc; the same argument in the second circle fixes ∠AQB. In triangle PBQ the angles at P and at Q are therefore both constant, so the third angle ∠PBQ is constant too — indeed the triangle keeps a fixed SHAPE and only changes size. Because the size does change, the length PQ, the height from B and the area all vary; only the angle is locked. (Rotate the line until it is perpendicular to AB in one circle and PQ visibly stretches.)