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  1. #1
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    Prove this!

    In a triangle ABC, BD and CE are the medians of a triangle, meet at centroid G.
    Prove that BG=2GD.
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  2. #2
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    Centroid of a triangle

    Hello snigdha
    Quote Originally Posted by snigdha View Post
    In a triangle ABC, BD and CE are the medians of a triangle, meet at centroid G.
    Prove that BG=2GD.
    If you understand how you can use vectors in geometry, you'll find a vector proof just here.

    But if you want a more 'traditional' proof, look at the diagram I've attached.

    In the diagram, $\displaystyle E$ and $\displaystyle D$ are the mid-points of $\displaystyle AB$ and $\displaystyle AC$ respectively.

    If we consider the area of $\displaystyle \triangle ABC$ when its base is $\displaystyle AB$, we see that:
    area of $\displaystyle \triangle AEC = \tfrac12$ area of $\displaystyle \triangle ABC$
    because the base ($\displaystyle AE$) of $\displaystyle \triangle AEC$ is one-half of the base ($\displaystyle AB$) of $\displaystyle \triangle ABC$, and the height of each triangle is the same.

    Similarly, when we consider $\displaystyle AC$ as the base of $\displaystyle \triangle ABC$, we get:
    $\displaystyle \triangle ABD = \tfrac12 \triangle ABC$
    Therefore:
    $\displaystyle \triangle AEC = \triangle ABD = \tfrac12 \triangle ABC$
    With the colours I've used in the diagram, this is:
    blue area + green area + yellow area = red area + yellow area + green area
    So, if we subtract the common area - the quadrilateral $\displaystyle AEGD$ (yellow area + green area) - from each triangle, we get:
    $\displaystyle \triangle BGE = \triangle CGD$
    In colours:
    red area = blue area
    But we can also see that, because $\displaystyle \triangle BGE$ (red) and $\displaystyle \triangle AGE$ (yellow) have equal bases and the same height:
    $\displaystyle \triangle BGE=\triangle AGE$
    In colours:
    red area = yellow area
    Similarly:
    $\displaystyle \triangle AGD=\triangle CGD$
    In colours:
    green area = blue area
    Thus all four coloured triangles have the same area. Therefore
    $\displaystyle \triangle AGB = \tfrac23\triangle ABD$
    But these triangles have a common base $\displaystyle AB$. Therefore the height of $\displaystyle \triangle AGB = \tfrac23$ of the height of $\displaystyle \triangle ABD$. So, by similar triangles:
    $\displaystyle BG = \tfrac23BD$

    $\displaystyle \Rightarrow BG = 2 GD$
    Tricky, isn't it? (It's much easier to use the vector method!)

    Grandad
    Attached Thumbnails Attached Thumbnails Prove this!-untitled.jpg  
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  3. #3
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    Quote Originally Posted by snigdha View Post
    In a triangle ABC, BD and CE are the medians of a triangle, meet at centroid G.
    Prove that BG=2GD.
    hi

    first , prove that $\displaystyle \triangle AED$ is similar to $\displaystyle \triangle ABC$

    Proof : $\displaystyle \angle A$ is a common angle for both triangles . We also see that AB=2AE and AC=2AD . Since the ratio of 2 corresponding sides are equal and the included angles are equal , proved then .

    so $\displaystyle \frac{AE}{AB}=\frac{ED}{BC}$ ----1

    Then , prove that $\displaystyle \triangle EDG $ is similar to $\displaystyle \triangle CBG$ .

    Proof : Since $\displaystyle \angle AED =\angle ABC $ and $\displaystyle \angle ADE=\angle ACB$ , ED is parallel to BC .

    hence , $\displaystyle \angle DEG=\angle BCE$ , $\displaystyle \angle EDG=\angle CBD$ (alternate angles) , and $\displaystyle \angle EGD=\angle BGC$ (opposite angles)

    since , all corresponding angles are equal , hence proved .

    so $\displaystyle \frac{DG}{BG}=\frac{ED}{BC}$ ---2

    From 1 , since $\displaystyle \frac{AE}{AB}=\frac{1}{2}$ , so $\displaystyle \frac{ED}{BC}=\frac{1}{2}$ , and it follows that $\displaystyle \frac{DG}{BG}=\frac{1}{2}$
    so BG=2GD
    Attached Thumbnails Attached Thumbnails Prove this!-triangle-median.bmp  
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