Lesson One property of a median in a triangle

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<H2>One property of a median in a triangle</H2>

<H3>Problem 1</H3>In a triangle, &nbsp;a median drawn from the vertex to the base bisects every segment parallel to the base and connecting two other sides. &nbsp;Prove.

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Solution

Let &nbsp;ABC&nbsp; be a triangle and &nbsp;CD&nbsp; be a median drawn from the vertex &nbsp;C&nbsp; to the side &nbsp;AB&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

(Figure 1). &nbsp;The statement is that if &nbsp;EF&nbsp; is a straight segment connecting the sides

&nbsp;AC&nbsp; and &nbsp;BC&nbsp; and parallel to the side &nbsp;AB&nbsp; then the median &nbsp;CD&nbsp; bisects &nbsp;EF.

Let &nbsp;EF&nbsp; be such a segment, &nbsp;and let &nbsp;G&nbsp; be the intersection point of the segment &nbsp;EF&nbsp;

with the median &nbsp;CD. &nbsp;

Then the triangle &nbsp;{{{DELTA}}}&nbsp;ECG&nbsp; is similar to the triangle &nbsp;{{{DELTA}}}&nbsp;ACD&nbsp; according with the

Theorem 1&nbsp; of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/In-a-triangle-a-straight-line-parallel-to-its-side-cuts-off-a-similar-triangle.lesson>In a triangle a straight line parallel to its side cuts off

a similar triangle</A>&nbsp; in this site. &nbsp;Hence, &nbsp;the corresponding sides of these triangles are

proportional:

{{{abs(EG)/abs(AD)}}} = {{{abs(CG)/abs(CD)}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

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{{{drawing( 220, 220, 0.5, 5.5, 0, 5.5, 
 line( 1.0, 1.0, 5.0, 1.0), 
 line( 1.0, 1.0, 4.0, 5.0),
 line( 4.0, 5.0, 5.0, 1.0),

locate(1.0, 1.0, A),
            locate(5.0, 1.0, B),
            locate(4.0, 5.4, C),

green(line(4, 5, 3.0, 1)),
            locate(2.9, 1, D),

green(line(1.75, 2, 4.75, 2)),
            locate(1.50, 2.3, E),
            locate(4.80, 2.3, F),

locate(3.25, 2.0, G)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. &nbsp;To the Problem 1

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

{{{abs(GG)/abs(BD)}}} = {{{abs(CG)/abs(CD)}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

The proportions &nbsp;(1)&nbsp; and &nbsp;(2)&nbsp; imply that

{{{abs(EG)/abs(AD)}}} = {{{abs(FG)/abs(BD)}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)

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

This is what has to be proved.

My other lessons on similar triangles in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similar-triangles.lesson>Similar triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similarity-tests-for-triangles.lesson>Similarity tests for triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Proofs-of-Similarity-tests-for-triangles.lesson>Proofs of Similarity tests for triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/In-a-triangle-a-straight-line-parallel-to-its-side-cuts-off-a-similar-triangle.lesson>In a triangle a straight line parallel to its side cuts off a similar triangle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-similar-triangles.lesson>Problems on similar triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similarity-tests-for-right-angled-triangles.lesson>Similarity tests for right-angled triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-similarity-for-right-angled-triangles.lesson>Problems on similarity for right-angled triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-similarity-for-right-angled-and-acute-triangles.lesson>Problems on similarity for right-angled and acute triangles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/One-property-of-a-trapezoid.lesson>One property of a trapezoid</A> &nbsp;&nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/Miscellaneous-problems-on-similar-triangles.lesson>Miscellaneous problems on similar triangles</A>

under the current topic, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-similar-triangles.lesson>Solved problems on similar triangles</A>

under the topic &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.