Possibly Unsolvable Circle Geometry Question

Question

I was recently trying to brush up on my circle geometry when I came across the below question:

A triangle PQR is inscribed in a circle, with a point W on the arc of QR. Furthermore, WX is perpendicular to PR produced, WZ is perpendicular to PQ and WY is perpendicular to RQ.

ii) Having proven that WXRY and WYZQ are cyclic quadrilaterals prove that W, Y, Z are collinear.

I attempted it yet was unable to solve it. However, furthermore the solutions were illogical. I’m not sure if the question is genuinely impossible and that the creators had made a mistake (judging by the incorrect solutions) or that there is a method to working it out.

Answer 1

Since WX $\perp$ XP and WY ⊥ RQ, WXRY is cyclic. Then,

$\angle XYW = \angle XRW = 180 - \angle PRW\tag{1}$

Similar, since WZ $\perp$ QP and WY ⊥ RQ, WYZQ is cyclic. Then,

$\angle WYZ = 180 - \angle PQW\tag{2}$

Moreover, since RPQW is cyclic, we have $\angle PRW + \angle PQW = 180$. From (1) and (2),

$$\angle WYZ + \angle XYW = 180$$

Thus, X, Y, and Z are collinear.

Answer 2

The question is correctly stated, and here is a hint:

Connect WP, XZ, RW, XY. Notice that WXRY and XWZP are cyclic quadrilaterals. Now try to find the relations between angles. Ask yourself: what does it mean if WXRY and XWZP are cyclic quadrilaterals?