Lesson One property of a median in a triangle

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One property of a median in a triangle


Problem 1

In a triangle,  a median drawn from the vertex to the base bisects every segment parallel to the base and connecting two other sides.  Prove.

Solution

Let  ABC  be a triangle and  CD  be a median drawn from the vertex  C  to the side  AB                    
(Figure 1).  The statement is that if  EF  is a straight segment connecting the sides
 AC  and  BC  and parallel to the side  AB  then the median  CD  bisects  EF.

Let  EF  be such a segment,  and let  G  be the intersection point of the segment  EF 
with the median  CD.  

Then the triangle  DELTA ECG  is similar to the triangle  DELTA ACD  according with the
Theorem 1  of the lesson  In a triangle a straight line parallel to its side cuts off
a similar triangle  in this site.  Hence,  the corresponding sides of these triangles are
proportional:

abs%28EG%29%2Fabs%28AD%29 = abs%28CG%29%2Fabs%28CD%29.            (1)


    Figure 1.  To the Problem 1

The triangle  DELTA FCG  is similar to the triangle  DELTA BCD  by the same reason.  It gives the proportion of their corresponding sides

abs%28GG%29%2Fabs%28BD%29 = abs%28CG%29%2Fabs%28CD%29.            (2)

The proportions  (1)  and  (2)  imply that

abs%28EG%29%2Fabs%28AD%29 = abs%28FG%29%2Fabs%28BD%29.            (3)

Since  CD  is the median in the triangle  ABC,  we have  |AD| = |BD|.  Therefore,  |EG| = |FG|  from the proportion  (3).

This is what has to be proved.


My other lessons on similar triangles in this site are
    - Similar triangles,
    - Similarity tests for triangles,
    - Proofs of Similarity tests for triangles,
    - In a triangle a straight line parallel to its side cuts off a similar triangle,
    - Problems on similar triangles,
    - Similarity tests for right-angled triangles,
    - Problems on similarity for right-angled triangles,
    - Problems on similarity for right-angled and acute triangles,
    - One property of a trapezoid   and
    - Miscellaneous problems on similar triangles
under the current topic,  and
    - Solved problems on similar triangles
under the topic  Geometry  of the section  Word problems.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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