prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and has length half that of the third side
Proof: Consider triangle ABC.
Join mid-points of AB and AC.
Let mid-point of AB be M and mid-point of BC be N.
Now we have two triangles ABC and AMN
In triangle ABC and triangle AMN
angle BAC=angle MAN…as A-M-B and A-N-C.
thus triangle ABC∼ triangle AMN.
Thus angle AMN=angle ABC as triangles are similar
and MN ∥ BC as corresponding angles are equal.
Also MN/BC=AM/AB=1/2 (sides of similar triangle.)