SOLUTION: R, S, and T are, respectively, the midpoints of the sides AB, AD, and CD of a square. Prove that DR and ST trisect each other.

Question 455308: R, S, and T are, respectively, the midpoints of the sides AB, AD, and CD of a square. Prove that DR and ST trisect each other.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!

I presume your question is "Prove that DR and ST split the other segment in a ratio of 1:2" (since "trisect means "to split into three equal parts").

Let P be the intersection of DR and ST. One can show, using AAA similarity, that triangles SPD and TPR are similar. Also, TR:SD = 2:1, so we can let TR = s and SD = s/2. Since the ratios among similar triangles are equal, this implies that TP:SP = 2:1 and RP:QP = 2:1. This means that we can let TP = 2k, SP = k for some k and note that TP/TS = TP/(TP + SP) = 2k/(2k + k) = 2/3, and the same symmetry applies to the other lengths.
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