Question 455308
{{{drawing(300,300,0,10,0,10,


line(2,2,8,2),
line(8,2,8,8),
line(8,8,2,8),
line(2,8,2,2),
locate(2,2,A),
locate(8,2,B),
locate(8,8,C),
locate(2,8,D),
locate(5,2,R),
locate(2,5,S),
locate(5,8,T),
line(2,8,5,2),
line(2,5,5,8),
line(5,2,5,8)
)
}}}


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.