Geometry
# Congruent and Similar Triangles

In the above diagram, \(\angle ABD = \angle BCE = \angle CAF.\) Given the lengths \[\lvert \overline{AB}\rvert=12, \lvert \overline{AC}\rvert =7, \lvert \overline{BC}\rvert = 14,\] what is \(\lvert \overline{DE}\rvert : \lvert \overline{EF}\rvert : \lvert \overline{DF}\rvert?\)

**Note:** The above diagram is not drawn to scale.

In the above quadrilateral \(ABCD,\) \(\overline{AD} \parallel \overline{EF} \parallel \overline{BC}, \) and

\[ \lvert\overline{AE}\rvert = 2\lvert\overline{EB}\rvert, \lvert\overline{BM}\rvert = 4, \lvert\overline{AD}\rvert = 6, \lvert\overline{BC}\rvert = 9,\] where \(\lvert\overline{AE}\rvert\) denotes the length of \(\overline{AE}.\) What is \(\lvert\overline{DO}\rvert?\)

**Note:** The above diagram is not drawn to scale.

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