Geographic Profiling Using Cyclic

[dekoi]

How does one find the most probable central location of something/someone when four points of their earlier location have been recorded and drafted? For example, if one is given points A, B, D, and C, how would they find the central location? This method must use cyclic quadrilaterals, and four circles which are each circumscribed by a different variation of three points.


````````````````A

B````````````````````````````C







`````````D


Perhaps the cyclic quadrilateral created by all the circumscribed circles' centers narrows down the location to inside the quadrilateral? Therefore, if one created a cyclic quad which touches all four origins, would the location be inside that cyclic quad? Maybe after drawing this cyclic quadrilateral which touches all four centers of the circles, one would connect opposite vertices, and the poitn at which both lines intersect, is the exact location.

However, that would not be using the cyclic quadrilateral created by the original points, A, B, C, and D.

 

[dekoi]

Ok, i have figured out that if you were to draw perpendicular bisectors to the initial quadrilateral created by the initial points, A, B, C, and D, you will creat an identical quadrilateral to the cyclic quadrilateral created with the four circles' centers.

Therefore, the quadrilateral created by the bisectors of the initial quadrilateral's sides is also the quadrilateral created with the centers of the circumscribed circles.


Would anyone happen to have an explanation for this?

 

[dekoi]

I have figured out why that identical quadrilateral appears in both situations. It is because of the Chord Right Bisector Property.

Therefore, i have basically solved the problem (or at least close to solving it). If anyone has any suggestions or disputes, please inform me.

One more question: How would one go about finding the center of a quadrilateral? Or do only cyclic quadrilaterals have centers? I would think so.