Analytic Proofs of Theorems Previously Proved by Synthetic Means
Let 𝐴(30,40), 𝐵(60,50), and 𝐶(75,120) be vertices of a triangle.
a. Find the coordinates of the midpoint 𝑀 of 𝐴𝐵 and the point 𝐺1 that is the point one-third of the way along 𝑀𝐶, closer to 𝑀 than to 𝐶.
b. Find the coordinates of the midpoint 𝑁 of 𝐵𝐶 and the point 𝐺2 that is the point one-third of the way along 𝑁𝐴, closer to 𝑁 than to 𝐴.
c. Find the coordinates of the midpoint 𝑅 of 𝐶𝐴 and the point 𝐺3 that is the point one-third of the way along 𝑅𝐵, closer to 𝑅 than to 𝐵.
a. Given triangle 𝐴𝐵𝐶 with vertices 𝐴(𝑎1, 𝑎2), 𝐵(𝑏1, 𝑏2), and 𝐶(𝑐1, 𝑐2), find the coordinates of the point of
concurrency of the medians.
b. Let 𝐴(−23, 12), 𝐵(13, 36), and 𝐶(23,−1) be vertices of a triangle. Where will the medians of this triangle intersect? (Use “Tyler’s formula” from part (a) to complete this problem.
Prove that the diagonals of a parallelogram bisect each other)
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