## To prove the Theorem of Apollonius using analytic methods, we can use the coordinates of the points in triangle ABC. Let's assume that point A has coordinates (x1, y1), point B has coordinates (x2, y2), and point C has coordinates (x3, y3).

1. Calculate the coordinates of point M, the midpoint of BC:

- The x-coordinate of point M is the average of the x-coordinates of points B and C, so: xM = (x2 + x3) / 2.

- The y-coordinate of point M is the average of the y-coordinates of points B and C, so: yM = (y2 + y3) / 2.

2. Calculate the distances AB, AC, AM, and MC using the distance formula:

- Distance AB = β((x2 - x1)^2 + (y2 - y1)^2)

- Distance AC = β((x3 - x1)^2 + (y3 - y1)^2)

- Distance AM = β((xM - x1)^2 + (yM - y1)^2)

- Distance MC = β((xM - x3)^2 + (yM - y3)^2)

3. Square both sides of the equation to get rid of the square roots:

- AB^2 = (x2 - x1)^2 + (y2 - y1)^2

- AC^2 = (x3 - x1)^2 + (y3 - y1)^2

- AM^2 = (xM - x1)^2 + (yM - y1)^2

- MC^2 = (xM - x3)^2 + (yM - y3)^2

4. Substitute the values of xM and yM from step 1 into the equations for AM^2 and MC^2:

- AM^2 = (xM - x1)^2 + (yM - y1)^2

= ((x2 + x3) / 2 - x1)^2 + ((y2 + y3) / 2 - y1)^2

- MC^2 = (xM - x3)^2 + (yM - y3)^2

= ((x2 + x3) / 2 - x3)^2 + ((y2 + y3) / 2 - y3)^2

5. Simplify the expressions for AM^2 and MC^2:

- AM^2 = ((x2 - x1) / 2)^2 + ((y2 - y1) / 2)^2

= (x2 - x1)^2 / 4 + (y2 - y1)^2 / 4

- MC^2 = ((x3 - x3) / 2)^2 + ((y3 - y3) / 2)^2

= (x2 - x3)^2 / 4 + (y2 - y3)^2 / 4

6. Multiply both AM^2 and MC^2 by 2:

- 2AM^2 = (x2 - x1)^2 / 2 + (y2 - y1)^2 / 2

- 2MC^2 = (x2 - x3)^2 / 2 + (y2 - y3)^2 / 2

7. Add the equations AB^2 + AC^2 = 2AM^2 + 2MC^2:

- AB^2 + AC^2 = (x2 - x1)^2 + (y2 - y1)^2 + (x3 - x1)^2 + (y3 - y1)^2

= (x2 - x1)^2 + (x3 - x1)^2 + (y2 - y1)^2 + (y3 - y1)^2

= (x2 - x1)^2 + (x1 - x3)^2 + (y2 - y1)^2 + (y1 - y3)^2

= (x2 - x1)^2 + (x3 - x1)^2 + (y2 - y1)^2 + (y3 - y1)^2

= 2[(x2 - x1)^2 + (y2 - y1)^2 / 2] + 2[(x2 - x3)^2 + (y2 - y3)^2 / 2]

= 2AM^2 + 2MC^2

Therefore, we have proven the Theorem of Apollonius using analytic methods.

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