Hey Thork, your Everest photo seems to indicate a round earth.
In this picture we can see two high mountain peaks. The closer one is Lhotse. The farther one is Kangchenjunga. Lhotse is 8,516 m high. Mount Everest (the camera height) is 8,848 m. The difference between these two heights is 332 meters.
Lets use some similar triangles.
Lhotse is 3.0 km away from the top of Mount Everest.
Mount Everest is 125 km away from Kangchenjunga.
332 m / 3km * 125km =13,800 meters.
This means that the, at the distance of Kangchenjunga from Mount Everest, on a flat earth, Lhotse should appear to be level with an altitude of 8,848-13,800 = -4950 meters at the distance of Kangchenjunga.
However, we see that Lhotse is about level with the top of Kangchenjunga (altitude 8,516 m). This cannot happen on a flat earth.
This is so wrong in so many ways I don't even know where to start. How did you dream this up? Are you saying the line of sight on a flat earth is less than that on a round earth. ? lol. Your diagram suggests so. Imagine if the earth curved. The mountain really would disappear and be lower. Are you saying the line of sight between two mountains at 8848 and 8516m is less than 125m on a flat earth? lol. On your diagram you suggest that the mountain on a flat earth would disappear under the horizon. ? Erm, no, that's what a round earth would do.
You seem to be critiquing the methodology, which is just fine. However, I did make a big mistake. The closer mountain is Makalu, (NOT Lhotse), which is 19.5 km away from everest and is 367 m lower than mount everest, at 8,481 m.
This means that the altitude level with apparently level with the top of Makalu at the distance of Kanchenjunga on a flat earth is:
367 m / 19.5km * 125km = 2.35 km lower than the height of everest, or 2.09 km lower than the height of Kanchenjunga.
Again, the methodology is just similar triangles, nothing wrong with it.
On a round earth, Kanchenjunga would appear to be lower than it would be on a flat earth. It would be lower by 1/2*d
2/R = 1/2*125
2/6380 = 1.22 km
This means that Makalu would appear to be level with a point .87 km lower than the summit on Kanchenjunga.
I will know draw your attention to Jannu (yes it's Jannu, you can check), the first peak a little bit to the right of Kanchenjunga. Its altitude is 7710 m and is 115 km away from everest.
Using the same methodology as above, Makalu should appear to be level with a point at Jannu 7710 - (8848-367*115/19.5) = 1030 meters lower than Jannu on a flat earth.
On a round earth, Makalu should appear to be level with a point 1000(1/2*115
2/6380) - 1030 = -6 meters below Janna. I.e. The two should be just level.
The top of Makalu is about level with Jannu, indicating a round earth.
Since the difference between the top of Kanchenjunga and Jannu is under a 1000 meters (870), we can say that 1000 meters is a visible difference in height at 115 km.
The top of Makalu is clearly about lined up with the top of Jannu rather than a point on Jannu 1000 meters below the summit, indicating that the earth is round
Edit:
This picture demonstrates the Makalu appears about level with Jannu, as RE predicts. If the earth were flat, Makalu would appear be well under Jannu.