Product of diagonals of a parallelogram
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I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are $v_1, v_2, v_3, v_4inmathbb R^2$. Let's say that the side $[v_1,v_4]$ is parallel to $[v_2,v_3]$. Is it true that
$||v_1-v_4||^2< ||v_1-v_3||cdot ||v_2-v_4||,$
that is,
the square of one side is less than the product of the diagonals?
I could also be just missing an obvious counterexample, but so far I can't prove it either.
geometry
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
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add a comment |
$begingroup$
I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are $v_1, v_2, v_3, v_4inmathbb R^2$. Let's say that the side $[v_1,v_4]$ is parallel to $[v_2,v_3]$. Is it true that
$||v_1-v_4||^2< ||v_1-v_3||cdot ||v_2-v_4||,$
that is,
the square of one side is less than the product of the diagonals?
I could also be just missing an obvious counterexample, but so far I can't prove it either.
geometry
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
$endgroup$
add a comment |
$begingroup$
I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are $v_1, v_2, v_3, v_4inmathbb R^2$. Let's say that the side $[v_1,v_4]$ is parallel to $[v_2,v_3]$. Is it true that
$||v_1-v_4||^2< ||v_1-v_3||cdot ||v_2-v_4||,$
that is,
the square of one side is less than the product of the diagonals?
I could also be just missing an obvious counterexample, but so far I can't prove it either.
geometry
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
$endgroup$
I came across a property which I am not sure if true in general. Suppose you have a parallelogram whose vertices are $v_1, v_2, v_3, v_4inmathbb R^2$. Let's say that the side $[v_1,v_4]$ is parallel to $[v_2,v_3]$. Is it true that
$||v_1-v_4||^2< ||v_1-v_3||cdot ||v_2-v_4||,$
that is,
the square of one side is less than the product of the diagonals?
I could also be just missing an obvious counterexample, but so far I can't prove it either.
geometry
geometry
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
asked Dec 10 '18 at 19:56
chhrochhro
1,242311
1,242311
add a comment |
add a comment |
1 Answer
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Consider a rhombus with unit sides which is very long and thin. As it gets
pointier, one diagonal tends to zero and the other to length $2$, so the product
of the diagonals gets very small.
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
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$begingroup$
Consider a rhombus with unit sides which is very long and thin. As it gets
pointier, one diagonal tends to zero and the other to length $2$, so the product
of the diagonals gets very small.
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
$endgroup$
add a comment |
$begingroup$
Consider a rhombus with unit sides which is very long and thin. As it gets
pointier, one diagonal tends to zero and the other to length $2$, so the product
of the diagonals gets very small.
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
$endgroup$
add a comment |
$begingroup$
Consider a rhombus with unit sides which is very long and thin. As it gets
pointier, one diagonal tends to zero and the other to length $2$, so the product
of the diagonals gets very small.
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
$endgroup$
Consider a rhombus with unit sides which is very long and thin. As it gets
pointier, one diagonal tends to zero and the other to length $2$, so the product
of the diagonals gets very small.
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
answered Dec 10 '18 at 20:00
Lord Shark the UnknownLord Shark the Unknown
104k1160132
104k1160132
add a comment |
add a comment |
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