The Theorum of Apollonius states in triangle ABC, if M is the midpoint of BC, then AB^2 + AC^2 = 2AM^2 + 2MC^2. Prove this theorum using analytic methods.
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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 = √((x2 - x1)^2 + (y2 - y1)^2)
- Distance AC = √((x3 - x1)^2 + (y3 - y1)^2)
- Distance AM = √((xM - x1)^2 + (yM - y1)^2)
- Distance MC = √((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.
If you would like to see a visual proof or further explanation, I recommend searching for "Theorem of Apollonius proof" on a search engine like Google for more detailed results and visual representations.