Two circles touch each other externally at the point C. A direct common tangent touches the first circle at P and the second at Q. The common tangent drawn at C meets PQ at R. Which statement is correct?

R is an external point of the first circle from which two tangents are drawn, touching at P and at C, so RP = RC. The same argument on the second circle gives RQ = RC. Hence RP = RC = RQ, so R is the midpoint of PQ and C lies on the circle whose diameter is PQ. An angle in a semicircle is a right angle, so ∠PCQ = 90°. The first option contradicts the equal-tangents theorem it should be using; the ratio option is what one would expect if R were on the line of centres instead; and the two radii to C are opposite rays, so the angle between them is 180°, not 90°.