Triangle Median Theorem proof
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Does anyone know what theorem this is? Or how to prove it?
Let $ABC$ be a triangle with sides $a, b, c$. Let $D$ be the midpoint of side $BC$ and $w = AD$ the length of the median through $A$. Prove that:
$$w^2={b^2over 2}+{c^2over 2}-{a^2over 4}.$$
geometry triangle
edited Nov 22 at 13:23
greedoid
36.5k114592
asked Nov 22 at 12:54
Matlab rookie
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up vote
1
down vote
favorite
Does anyone know what theorem this is? Or how to prove it?
Let $ABC$ be a triangle with sides $a, b, c$. Let $D$ be the midpoint of side $BC$ and $w = AD$ the length of the median through $A$. Prove that:
$$w^2={b^2over 2}+{c^2over 2}-{a^2over 4}.$$
geometry triangle
edited Nov 22 at 13:23
greedoid
36.5k114592
asked Nov 22 at 12:54
Matlab rookie
307
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Does anyone know what theorem this is? Or how to prove it?
Let $ABC$ be a triangle with sides $a, b, c$. Let $D$ be the midpoint of side $BC$ and $w = AD$ the length of the median through $A$. Prove that:
$$w^2={b^2over 2}+{c^2over 2}-{a^2over 4}.$$
geometry triangle
edited Nov 22 at 13:23
greedoid
36.5k114592
asked Nov 22 at 12:54
Matlab rookie
307
Does anyone know what theorem this is? Or how to prove it?
Let $ABC$ be a triangle with sides $a, b, c$. Let $D$ be the midpoint of side $BC$ and $w = AD$ the length of the median through $A$. Prove that:
$$w^2={b^2over 2}+{c^2over 2}-{a^2over 4}.$$
geometry triangle
geometry triangle
edited Nov 22 at 13:23
greedoid
36.5k114592
asked Nov 22 at 12:54
Matlab rookie
307
edited Nov 22 at 13:23
greedoid
36.5k114592
asked Nov 22 at 12:54
Matlab rookie
307
edited Nov 22 at 13:23
greedoid
36.5k114592
edited Nov 22 at 13:23
greedoid
36.5k114592
edited Nov 22 at 13:23
greedoid
36.5k114592
36.5k114592
asked Nov 22 at 12:54
Matlab rookie
307
asked Nov 22 at 12:54
Matlab rookie
307
asked Nov 22 at 12:54
Matlab rookie
307
307
add a comment |
add a comment |
1 Answer
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Reflect $A$ across $D$, we get new point $E$ such that $ABEC$ is paralelogram.
By paralelogram identity (which can be easyl proved say by cosine law) we have $$2AB^2+2AC^2 = AE^2+BC^2$$
$$2c^2+2b^2 = a^2+(2w)^2$$
Now you can finish...
answered Nov 22 at 13:20
greedoid
36.5k114592
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Reflect $A$ across $D$, we get new point $E$ such that $ABEC$ is paralelogram.
By paralelogram identity (which can be easyl proved say by cosine law) we have $$2AB^2+2AC^2 = AE^2+BC^2$$
$$2c^2+2b^2 = a^2+(2w)^2$$
Now you can finish...
answered Nov 22 at 13:20
greedoid
36.5k114592
add a comment |
up vote
1
down vote
accepted
Reflect $A$ across $D$, we get new point $E$ such that $ABEC$ is paralelogram.
By paralelogram identity (which can be easyl proved say by cosine law) we have $$2AB^2+2AC^2 = AE^2+BC^2$$
$$2c^2+2b^2 = a^2+(2w)^2$$
Now you can finish...
answered Nov 22 at 13:20
greedoid
36.5k114592
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Reflect $A$ across $D$, we get new point $E$ such that $ABEC$ is paralelogram.
By paralelogram identity (which can be easyl proved say by cosine law) we have $$2AB^2+2AC^2 = AE^2+BC^2$$
$$2c^2+2b^2 = a^2+(2w)^2$$
Now you can finish...
answered Nov 22 at 13:20
greedoid
36.5k114592
Reflect $A$ across $D$, we get new point $E$ such that $ABEC$ is paralelogram.
By paralelogram identity (which can be easyl proved say by cosine law) we have $$2AB^2+2AC^2 = AE^2+BC^2$$
$$2c^2+2b^2 = a^2+(2w)^2$$
Now you can finish...
answered Nov 22 at 13:20
greedoid
36.5k114592
edited Nov 22 at 13:37
answered Nov 22 at 13:20
greedoid
36.5k114592
answered Nov 22 at 13:20
greedoid
36.5k114592
answered Nov 22 at 13:20
greedoid
36.5k114592
36.5k114592
add a comment |
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