- Hi, all.

My conjecture is the following :

Let P be in the interior of triangle ABC, and let the lines AP, BP,

CP intersect the sides BC, CA, AB in L, M, N, respectively.

Then, 3/2 >= MN/BC + NL/CA + LM/AB ??

Is this already known result?

Is this true?

Yours sincerely,

Hojoo Lee - Dear Hojoo,

you wrote :

> My conjecture is the following :

This is obviously false :

>

> Let P be in the interior of triangle ABC, and let the lines AP, BP,

> CP intersect the sides BC, CA, AB in L, M, N, respectively.

> Then, 3/2 >= MN/BC + NL/CA + LM/AB ??

>

> Is this already known result?

> Is this true?

>

take (BC,CA,AB)=(13,9,6) and P on the median AA'.

Let f=MN/BC + NL/CA + LM/AB

When P->A, f->5sqrt(65)/36 = 1.1...

When P->A', f->101/36 = 2.8...

Now, my question : since f is bounded inside the triangle, what are its

limit sup and limit inf ?

Best regards

Bernard - --- In Hyacinthos@y..., insight_love@h... wrote:
> Hi, all.

If the triangle ABC is acute-angled and P is its orthocenter, then

>

> My conjecture is the following :

>

> Let P be in the interior of triangle ABC, and let the lines AP, BP,

> CP intersect the sides BC, CA, AB in L, M, N, respectively.

> Then, 3/2 >= MN/BC + NL/CA + LM/AB ??

>

> Is this already known result?

> Is this true?

>

> Yours sincerely,

> Hojoo Lee

the inequality is true:

MN/BC = cosA, NL/CA=cosB,LM/AB=cosC and

cosA + cosB + cosC <=3/2 (a well-known inequality)

Achilleas Sinefakopoulos - If we take A=(0,0); B=(0,1) C=(n,1)

L (in segment BC) close enough to B.

M=(m2,m2) (in segment AC) with m1=1

It is easy to see( taking n big enough) that the function

MN/BC + NL/CA + LM/AB

take values as small as we want.

manuel

> > Let P be in the interior of triangle ABC, and let the lines AP,

BP,

> > CP intersect the sides BC, CA, AB in L, M, N, respectively.

its

> > Then, 3/2 >= MN/BC + NL/CA + LM/AB ??

> >

> > Is this already known result?

> > Is this true?

> >

> This is obviously false :

> take (BC,CA,AB)=(13,9,6) and P on the median AA'.

>

> Let f=MN/BC + NL/CA + LM/AB

>

> When P->A, f->5sqrt(65)/36 = 1.1...

> When P->A', f->101/36 = 2.8...

>

> Now, my question : since f is bounded inside the triangle, what are

> limit sup and limit inf ?