Prove triangle theorems including angle sum, isosceles base angles, midsegment theorem, and medians concurrence.
Find named centroid subsegments and the whole median
Problem
On median \(AM\), centroid \(G\) lies between \(A\) and \(M\) and \(AG~=~10\). Use \(AG:GM~=~2:1\) to calculate \(GM\), then add the \(\text{two}~\text{subsegments}\) to calculate the whole median \(AM\).
The centroid ratio is directional: the vertex-to-centroid piece has two parts while the centroid-to-midpoint piece has one. We’ll preserve the point order, use the known longer piece to find one ratio part, then add the adjacent pieces for the whole median and check the ordered ratio.
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