1. ## Geometric Construction #2

Given three medians of a triangle ABC, construct the triangle.

2. Originally Posted by MATNTRNG
Given three medians of a triangle ABC, construct the triangle.
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Given lines AO, BO, CO, you want to locate the positions of A, B, C on those lines so that they form the vertices of a triangle having those lines as medians. The size of the triangle is not determined by that condition, so you can fix A arbitrarily on the line AO. Construct the point D on the line AO, on the opposite side of O from A, so that $\displaystyle DO = \tfrac12AO$. Then D will lie on the side BC of the triangle.

The centroid O of the triangle is the centre of mass of three equal masses at the vertices. So the vertices A and C must be equidistant from the median BO. Construct a line AXY through A, perpendicular to BO, meeting BO at X, and such that AX = XY. Then construct a line through Y perpendicular to AXY (and therefore parallel to BOX). This line meets CO at the vertex C of the triangle, and thus determines the position of C.

Finally, you can construct B as the point where the line CD meets the median BOX.