A purposefully obtuse Euclidean geometry riddle from an old interactive fiction game

Question

I've been playing Praser 5, an old interactive fiction game (like Zork) made a few decades ago, and available here. In it, you wander from area to area solving riddles posed by mythical animals. Many of the puzzles involve lateral thinking. I've had quite a bit of difficulty with the following puzzle:

The sphinx stretches lazily and says, "Consider a Euclidean plane. Draw points O and S. Put I between them so that half the distance between O and I is the distance between O and the point one-third of the way from O to S. Say that S and I are one bloit apart. Now put D as far from O as S is from the point between I and O, so that the ratio of the distance between D and I and the distance between O and the point between D and I is twice the distance between O and the point between D and O. Mark E so that the distance between O and the point between E and the point between D and E is half the distance between O and I, and half the distance between S and E is the distance between the point between S and D and D. Now, P is placed so that the sum of the distances between P and S and between P and D, one of which is two bloits, is as long as E is far from the point three times as far from O as it is from D so that D is between that point and E. P is closer to I than it is to S, but if it weren't, H would be the point between P and D as far from where P really is as I is from where P would be. R is as far from H as O is from S and as far from D as D is from E. So what is the shape of the figure OSRHP, and what is its area in square bloits?" It smiles gently and curls up again.

So far, I have O, I, and S on a line, with O and I two bloits apart and S one bloit past I. D is two bloits above O, and E is two bloits below O.

Beyond that, I suspect that P is in the lower right, H in the upper middle, and R in the lower left. I think the result is a star/pentagram. However, to check your answer in game, you must simultaneously enter the shape and area.

Is my solution (a star) correct? And he can I find the area?

(Note: to try speaking to the Sphinx in-game, type arlumere).

Answer 1

Consider a Euclidean plane. Draw points O and S.

Put I between them so that half the distance between O and I is the distance between O and the point one-third of the way from O to S. Say that S and I are one bloit apart.

So I lies at the 1/3rd point between S and O.

Now put D as far from O as S is from the point between I and O, so that the ratio of the distance between D and I and the distance between O and the point between D and I is twice the distance between O and the point between D and O.

D is two bloits from O, and happens to form a right triangle which satisfies the other requirements.

Mark E so that the distance between O and the point between E and the point between D and E is half the distance between O and I, and half the distance between S and E is the distance between the point between S and D and D.

Putting E two bloits above O works here.

Now, P is placed so that the sum of the distances between P and S and between P and D, one of which is two bloits, is as long as E is far from the point three times as far from O as it is from D so that D is between that point and E.

"that point" is collinear with D and E. It's three bloits from O and one bloit from D. E is five bloits from "that point", so the two distances in the first clause are two and three bloits, although we don't know which is which. So P can be in four potential places, represented here as the intersection of the red and blue circles.

P is closer to I than it is to S, but if it weren't, H would be the point between P and D as far from where P really is as I is from where P would be.

The only candidate P closer to S is the one farthest to the right. We call this "Fake P". I is sqrt(5) bloits away from Fake P. We'll draw circles of radius sqrt(5) around each of our other candidate P's to see if any of them intersect the line between D and Fake P. This is where H is.

Only one of the circles intersects with the line. That circle belongs to the candidate P that lies on top of O.

R is as far from H as O is from S and as far from D as D is from E.

Only one point is 3 units from H and 4 units from D.

So what is the shape of the figure OSRHP, and what is its area in square bloits?

The shape is a parallelogram and its area is 6 square bloits.