I'm not saying that he already owned that hat. He could've constructed it from paper. But I'm not saying he didn't either.

What can we say about triangle PST?

Look at line ST. It's a tangent line, so it intersects the circle at one point. Call that point U. We don't know the length of SU or TU. We don't know if they are congruent to each other or not. But we do know one thing.

SU and SQ are two tangents to the circle from the same point, and they have the same length. Likewise, TU and TR have the same length. This is enough information to tell us something about the perimeter of the triangle:

PS + SU + UT + TP = PS + SQ + RT + TP = (PS + SQ) + (RT + TP) = PQ + PR = 2 PQ = 2 PR

The perimeter of the triangle is equal to the sum of the lengths of the two larger tangent lines. And because those tangents are equal to each other, we know that the perimeter equals twice the length of one of the tangents.

Also, the triangle could like a clown hat.