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Question
Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side
Solution
In ΔABC
`vec"OA" = vec"a"`
`vec"OB" = vec"b"` and
`vec"OC" = vec"c"`
D = midpoint of AB = `vec"OD" = (vec"a" + vec"b")/2`
E = midpoint of AB = `vec"OE" = (vec"a" + vec"c")/2`
Now `vec"DE" = vec"OE" - vec"D" = (vec"a" + vec"c")/2 - (vec"a" + vec"b")/2`
= `(vec"a" + vec"c" - vec"a" + vec"b")/2`
= `(vec"c" - vec"b")/2`
= `vec"BC"/2`
`vec"DE" = vec"BC"/2`
⇒ `vec"DE" || "to" vec"BC"` and half of BC.
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