Calculating middle point based on three corners and distances (Trilateration)

Also known as detecting your current geoposition based on three 5G towers.

The Problem

Given the three 2D points A, B and C

and the distance/timing to the points from unkown origin point (P)

determine the P's coordinates.

Solution

Given: Triangle

Given: Distances to the fourth point. If the timing in given, you can just divide by the speed of radio.

Step 1: Distance between A and B

Step 2: Directional vector from A to B

Step 3: ???

Step 4: Triangle height

Step 5: Intersection center

Step 6: Intersection points between two circles

Step 7: Repeat for BC and CA

In the end you'll have three pairs of intersection points. If the given distances have the same origins, then the circles will also intersect in the same spot.

Step 8: Deploy a tactical AC air strike

Step 9: Help Amon write a proper explanation

Desmos demonstration

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