Perpendicular Lines

AD, BE and CF...

Question

AD, BE and CF are three concurrent lines in a triangle ABC meeting the opposite sides in D, E and F respectively. Show that the joins of the midpoints of BC, CA and AB to the midpoints of AD, BE and CF are concurrent.

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Solution

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△ A B C

D, E, F

B C, C A, A B

¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ B E + ¯ ¯¯¯¯¯¯ ¯ C F =

A B C, D, E

F

B C, C A

A B

¯ ¯¯¯¯¯¯¯ ¯ A D + ¯ ¯¯¯¯¯¯ ¯ B E + ¯ ¯¯¯¯¯¯ ¯ C F = ¯ ¯ ¯ 0

D E F

B C, C A, A B

Δ A B C

A D, B E

\frac{O D}{A D} + \frac{O E}{B E} + \frac{O F}{F C} = 1

A B C, D, E, F

B C, C A

A B

\to A D = - (\to B E + \to C F)

\vec{A D} + \vec{B E} + \vec{C F} .

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