# How to find out if three points on Earth are on the same great circle arc?

I am trying to figure out whether 3 particular points on Earth's surface are on the same arc (of a great circle) or not. Using Google Earth's (also ArcGIS Earth's) line tool, I see that the points are approximately on the same arc as the one in the middle is only 3 km off where the distance between the neighboring points are more than 1000 km each.

Considering the oblate spheroid shape of Earth, can I confidently conclude from Google (and ArcGIS) Earth measurements that these points are not exactly on the same arc? How probable is it that approximations or some other factors cause 3 km deviation in 1000 km? Is there any method or tool that can help reaching to exact results?

• Through any three points on the surface of an ellipsoid, you can draw a circle. Therefore, these three points are on the same arc.
– KVO
Commented Jul 5, 2023 at 23:34
• Perhaps you meant the arc of a great circle, which lies in a plane passing through the center of the Earth.
– KVO
Commented Jul 5, 2023 at 23:38
• @KVO yes I mean the arc of a great circle, thanks for pointing it out. Commented Jul 6, 2023 at 7:10
• I was not aware that there is a SE community of GIS. I have asked the question there too. gis.stackexchange.com/questions/462900/… Commented Jul 6, 2023 at 7:27
• What you could do is to define the two great circles passing through two pairs of point, i.e., the one passing through A–B, and the one passing through B–C. Then you could compute the intersection points of these two great circles. Matlab has a function for that: mathworks.com/help/map/ref/gcxgc.html, you could write something similar in Python. The function will find the two intersection points between the two great circles (if so, then A–B–C are not on the same one) or will tell you that the pair of great circles are identical, meaning that the three points are on the same one. Commented Jul 6, 2023 at 7:36