In triangle $ABC,$ let $D,$ $E,$ $F$ be the midpoints of $\overline{BC},$ $\overline{AC},$ $\overline{AB},$ respectively.  Let $P,$ $Q,$ $R$ be the midpoints of $\overline{AD},$ $\overline{BE},$ $\overline{CF},$ respectively.  Compute
\[\frac{AQ^2 + AR^ 2+ BP^2 + BR^2 + CP^2 + CQ^2}{AB^2 + AC^2 + BC^2}.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
In triangle $ABC,$ let $D,$ $E,$ $F$ be the midpoints of $\overline{BC},$ $\overline{AC},$ $\overline{AB},$ respectively.  Let $P,$ $Q,$ $R$ be the midpoints of $\overline{AD},$ $\overline{BE},$ $\overline{CF},$ respectively.  Compute
\[\frac{AQ^2 + AR^ 2+ BP^2 + BR^2 + CP^2 + CQ^2}{AB^2 + AC^2 + BC^2}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{7}{8}
In triangle $ABC,$ let $D,$ $E,$ $F$ be the midpoints of $\overline{BC},$ $\overline{AC},$ $\overline{AB},$ respectively.  Let $P,$ $Q,$ $R$ be the midpoints of $\overline{AD},$ $\overline{BE},$ $\overline{CF},$ respectively.  Compute
\[\frac{AQ^2 + AR^ 2+ BP^2 + BR^2 + CP^2 + CQ^2}{AB^2 + AC^2 + BC^2}.\]
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