Random Points

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Mark Paxton

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I am not a student and have no mathematical knowledge but I am looking for an answer and maybe somebody can help.
The question is "What is the probability of six random points creating five isosceles triangles?"
I realise there must be some tolerance (say 1% or 0.5%) on the side lengths of the triangles, or some dimension to the points.
Thank you to anyone who has the time to consider this.
Mark
 
Thank you for your reply tkhunny. These points are on the Marlborough Downs in an area about 15 km square. Five of the points are located on the highest hill summits and no isosceles triangles are found between them. The sixth point is at the corner of all five isosceles triangles. It is marked on the ground by The Obelisk, Avebury's tallest Neolithic stone. I am trying to establish the probability of this stone being randomly located.
 
3rd: Are all the summits the same altitude? If not, then we're assuming isosceles from a satellite POV and that the Earth is flat?
4th: 0% Given a continuous distribution (with no masses), the probability of hitting a single point is zero (0%).

Probability Test:
If you drop three random points on a sphere, what is the probability that all three will land in or on the same hemisphere?

5th: From a practical POV, summits aren't single points and they move with geological events and obelisks aren't single points. Thus, exact isosceles triangles are hard to claim with 6 fat, moving targets. In other words, maybe they were relatively precise when the obelisk was built, but it is far less likely, some 4500ish years later.

Geometry Test:
Given five (5) non-colinear points, can one ALWAYS construct a central point that creates isosceles triangles with EVERY consecutive pair of points?
 
Thanks for your reply.
3rd. Yes the heights differ by about 40 meters. I have used Google Earth to make these measurements. The measuring tool offers "Map Length" and "Ground Length". There is occasionally a 0.1% difference between the two.

4th. I do not understand "a continuous distribution (with no masses)", is this referring to the need for geometric points and lines to have a dimension?
Probability Test. Is the equator between the two hemispheres fixed? If so I think the answer is a 3 to 1 probability they will land together in one of the two hemispheres. But if you could decide the equator after the points had fallen I think the probability of being able to isolate them in one hemisphere would be greater.

5th. Agreed that the target is "fat", but geologically stable. Hill summits on the Marlborough Downs relatively flat and are impossible to identify exactly, a small target area is defined on the summit area within which the precise geometric point is found. This gives a required tolerance of about 0.5% on the lengths of the isosceles legs.

Geometry Test: I really wish I knew the answer to your question. I am almost certain the answer is "no", but have no idea how to work it out.

Once again many thanks
 
That's pretty good. I was just trying to see if you were thinking straight.

Probability Test: If you presuppose the equator, which most people do, the probability is easy enough. The first lands anywhere, but the next two have to land in the right place, 1/2 * 1/2 = 1/4. However, if you let the first two points fall where they may, and then use them to establish an "equator", the third can land anywhere and fulfill the requirements, thus 100% chance. Your question of the fixed equator made me smile! Good work.

A continuous distribution carries the implication that your random selection can happen anywhere at all. There is not a place where things pile up or become more probable. There is no grid where points are allowed to fall. It is sometimes confusing to think that the probability of landing anywhere in particular is zero, but we know it landed somewhere - which sounds a lot like a non-zero probability.

The reason for the geometry test was to change the question a little. Not only should we be asking if there is a sixth point that can create all the isosceles triangles, we should also be asking if the first five points are so arranged that there even is a place for the sixth to fall to create the desired effect. Is this all sounding less and less probable? It should. However, I started at 0%, so can we get much lower? :)
 
Thank you for your reply.
The more I think about it the more certain I am that five random points will not always present the possibility of a sixth point being found that creates the corner of five isosceles triangles with the five points. Indeed the probability that such a point exists may be quite small. The picture shows the five triangles leading to this question.

Five triangles.jpg

The Obelisk is the northernmost point in the image, the other five points are hill summits. Four triangle sides share one line and two triangle sides share another. In both of these cases the geometric point located on the hill summit is fixed and the geometric point on The Obelisk is fixed. (The hill summit areas are determined with a combination of Google Earth's altitude readings, the OS Map 1:25000 and on sight observation. A geometric point is located within this area.) Measuring between these points using Google Earth's measuring tool the line length on the sides of the five isosceles triangles require a maximum tolerance of 0.5%. Put another way if all five triangle corners are located on The Obelisk the remaining corners are all found on the highest hill summit areas on the Downs.
I guess the root of my question is to ask: Would it be fair to say, in the context of high point geometry in the landscape, that the location of The Obelisk is unlikely to have been arrived at randomly? I think you are saying, strictly within this context, that the probability of finding such a point randomly is close to zero. Or indeed such a point may not necessarily always exist given a spread of only five random points. Am I close to the mark?
 
I see from your drawing that my assumption of the obelisk being the point between each pair of congruent sides is not correct. This complicates things a bit.

If my assumption were so, and with 5 points, there are 15 possible bases on which to build your triangles. Simply construct the Perpendicular Bisector on each base defined by each pair of points. These perpendicular bisectors must share a common point if one obelisk is to accomplish the task.

As my assumption is not so, there are 5 more sides to serve as a base of a possible isosceles triangle. Thus, I now enumerate 19 triangles to explore.

As a general rule, if we keep looking, we'll find something that seems interesting or mysterious. We might call this apophemia. Is there any significance to this relationship at all? We would have to ask the builders.
 
Excuse this late response to your reply, and thank you for it. I think the position of The Obelisk presents a dichotomy. On the one hand it could have been randomly chosen, on the other hand it could have been chosen because it is a unique point with regard to the geometry created with the surrounding hilltops. You initially suggested the probability of a point like this being found among six random points was 0%. This would suggest to me that it was not randomly chosen, being aware that preconceived beliefs about the ability of Neolithic people does not alter this statistical probability.

“Ask the builders” you suggest. This is possible in the sense that one can investigate the locations of ten other major Neolithic monuments in relation to regional high points.

All these locations are related geometrically to the high points, creating topographical isosceles triangles in a predictable manner. And again, in all cases, this geometry is statistically very unlikely to occur among a handful of random points. If this is the case is one not bound to conclude that these points on the ground, like that of The Obelisk, were not randomly located?
 
I have to agree that the position of the Obelisk was not randomly chose, given the fixed location of the other points.

I cannot say the same of the other points.
 
Again, thank you for this last reply. I'm unsure what you mean by "I cannot say the same of the other points."
 
I have no opinion and insufficient information to opine on the placement of the peaks.
 
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