Answers

There are many possible positions of Hive account@ahmadmangazap. However, for a participant to be considered as winner, they must have submitted at least one correct posiiton of Hive account@ahmadmangazap.

The easiest to get is (-4,10). Nevertheless, any participant who submitted any possible position of Hive account@ahmadmangazap in the original problem shall be considered as winner.

Solutions

Graphing method

We can visually obtain the position of Hive account@ahmadmangazap is to draw the points, and draw circles around Hive account@holovision with radius starting from 1 and increasing by 1, and Hive account@jfang003 with starting radius 2 and increasing by 2, until the circumferences of the circles touch each other at least once above 6 in the y-axis.

(12, 6.032) is a possible position of Hive account@ahmadmangazap based on the graph, though you have to use a big and precise graphing area (graphing apps such as Desmos is recommended).

Distance between A and H
= sqrt((10 - 12)^2 + (2 - 6.032)^2)
= sqrt((-2)^2 + (-4.032)^2)
= 4.5008

Distance between A and J
= sqrt((3 - 12)^2 + (6 - 6.032)^2)
= sqrt((-9)^2 + (0.032)^2)
= 9.0001

Because we used approximation in 6.032, we get rounding error in the distances. However, that is an acceptable error.

Midpoint method

Just place Hive account@ahmadmangazap somewhere on the graph such that Hive account@jfang003 becomes the midpoint of Hive account@ahmadmangazap and Hive account@holovision.

The midpoint formula is (xm, ym) = ((xa + xb)/2, (ya + yb)/2), such that:

(xm,ym) = Hive account@jfang003's position which is (3,6)
(xa, ya) = Hive account@ahmadmangazap's position which is unknown
(xb, yb) = Hive account@holovision's position which is (10,2)

Our equation is now (3,6) = ((xa + 10)/2, (ya + 2)/2). In the equation, xa = -4 and ya = 10. Therefore, a possible position of Hive account@ahmadmangazap is (-4,10).

Systemic method

Nevermind the solution below, since I believe I already explained enough using the easier solutions above, unless you want 🤯

This method involves systems of equations.

Let's use capital letters as positions of the persons. A for Hive account@ahmadmangazap, H for Hive account@holovision, and J for Hive account@jfang003.
Let (x,y) be the coordinates of Hive account@ahmadmangazap.

Let d be the distance of Hive account@ahmadmangazap from Hive account@holovision
Let 2d be the distance of Hive account@ahmadmangazap from Hive account@jfang003

We use the distance formula to get the distance between any two points. The distance formula is

the distance is equal to the square root of the sum of the squares of the difference between the change in y and change in x

Using the distance formula and the two distance equations above, we get the two equations:

d^2 = (10 - x)^2 + (2 - y)^2
d^2 = x^2 - 20x + 100 + y^2 - 4y + 4
Equation 1: d^2 = x^2 - 20x + y^2 - 4y + 104
(2d)^2 = (3 - x)^2 + (6 - y)^2
(2d)^2 = x^2 - 6x + 9 + y^2 - 12y + 36
Equation 2: 4d^2 = x^2 - 6x + y^2 - 12y + 45

Since the d on Equation 1 is the same as d on Equation 2, we can substitute Equation 1 to Equation 2.

x^2 - 6x + y^2 - 12y + 45 = 4 * (x^2 - 20x + y^2 - 4y + 104)
x^2 - 6x + y^2 - 12y + 45 = 4x^2 - 80x + 4y^2 - 16y + 416
3x^2 - 74x + 3y^2 - 4y + 371 = 0
x^2 - 74x/3 + y^2 - 4y/3 + 371/3 = 0
(x^2 - 74x/3 + 1369/9) + (y^2 - 4y/3 + 4/9) + 371/3 - (1369/9 + 4/9) = 0
(x + 37/3)^2 + (y - 2/3)^2 - 260/3 = 0
Equation 3: (x + 37/3)^2 + (y - 2/3)^2 = sqrt(260/3)^2

We can use any pair of x and y values satisfying Equation 3 for as long as y is greater than 6. The higher of Hive account@jfang003 and Hive account@holovision's positions has a y-intercept of 6.