First, you know that angle PQR is 90 degrees because PR is a diameter. You also know that angle RPQ is the same for both triangles. The questions then is, can we say that triangles PON and PRQ are similar?
We know that the hypotenuse of PON is a radius and the hypotenuse of PRQ is a diameter. Therefore the larger hypotenuse is twice the smaller. If the triangles are similar, their angles will be similar and their sides proportionate. Assume the length of ON is x and the length of PN is y. The area of PON is then 1/2 xy. If the triangles are similar, since the hypotenuse is double, all sides are double and RQ is 2x and PQ is 2y. That would make the area of PQR 1/2 (2x)*(2y) or 2XY. The area of the larger triangle would have to be four times the area of the smaller. Since the area of the quadrilateral ONRQ is three times the area of triangle PON, the area of the larger triangle is, in fact, four times the area of PON. That means the triangles are similar and angle PNO and PQR are both right angles, which makes the lines parallel.
Tani Wolff
