You are at a high spot at sunrise and point directly at the center of the sun.

Return to the same spot at noon and point again at the center of the sun.

This forms two lines in space. Where do those two lines cross?

Are they really two different lines?

Your location, tied to the dock?

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A high spot with an unobstructed view. Same spot both times.

Yes.

The earth continued its orbit around the sun between rise and noon; the two lines have the same origin (the sun) but not the same destination (the earthâ€™s position).
Hence, these theoretical lines will cross at the sunâ€™s center.

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What about the correction for semi-diameter?

They cross at the point where the person is standing.

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Agreed, depending on the observer. To an observer within the solar system the lines cross at the barycenter of the Sun/Earth. And, I guess, to an observer outside the solar system the probability of the lines intersecting at all is almost zero.

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Unfortunately, the barycenter of the system Sun+Earth is about 500 km off the sunâ€™s center, which is just nothing against the sunâ€™s radius of about 700 000 km.

Only the very distant four gas or ice giants, Jupiter to Neptune, are significant for the barycenter of the system Sun+AllPlanets.
Just now, Jupiter, Saturn and Neptune are nearly in line, and the solar systemâ€™s barycenter is 1.9 sun radii away from the sunâ€™s center.

However, the brainteaser is about two lines of sight of the sun from a fixed point on the (moving and rotating) earth, at different times.

Not arguing that. Iâ€™m just pointing out that the intersection, to an outside observer, would not be the first person observer. It would also not be the Sunâ€¦ exactly. It would be the barycenter of the two bodies (which as you correctly pointed out is within the Sun).

The original question was vague enough and teased with the word â€śbrainteaserâ€ť that a more detailed answer was appropriate.

An even more precise answer would be that the lines would intersect between the barycenter and the Sunâ€™s center of mass. Exactly where depends on the time of year and the first observerâ€™s latitude on Earth. Only twice a year would that be the barycenter.

Not intuitively obvious, at least it wasnâ€™t to me.

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The solar angle of this triangle is the equatorial parallax.
At a subsolar point, the centers of sun and earth and the viewer are in line, the angle is 0Â°.
At sun rise or set the angle is 8.794143 arcseconds, an IAU constant.

The nearby moon, with nearly the same angular diameter as the sun, has a much larger equatorial parallax, 0.9Â° to 1.03Â°, depending on the true distance earth<>moon.

0n the earthâ€™s orbit around the sun, the daily angular difference is about 1Â°.
Taking 6 hours between sunrise and noon, this position difference would be 15 arcminutes.

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Doesnâ€™t it depend on oneâ€™s frame of reference?

From Wikipedia Earthâ€™s Rotation

Itâ€™s the idea that the sun travels across the sky that is misleading in this case. If we use reference #3 in the graphic to define what we mean by â€śpoint at the sunâ€ť would it be correct to not consider the earthâ€™s orbit?

Is that a legit frame of reference? Seems like that would be the case if we use local apparent noon, which would have shifted a bit in six hours.