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Cevian triangles
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quasi
May 25, 2012, 5:16:31 PM5/25/12
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Conjecture:
If a triangle with integer side lengths a,b,c is such that,
for some interior point P of the triangle, the 3 Cevians
through P partition the triangle into 6 smaller triangles
with integer side lengths, then gcd(a,b,c) > 1.
quasi
If a triangle with integer side lengths a,b,c is such that,
for some interior point P of the triangle, the 3 Cevians
through P partition the triangle into 6 smaller triangles
with integer side lengths, then gcd(a,b,c) > 1.
quasi
1treePetrifiedForestLane
May 25, 2012, 5:38:27 PM5/25/12
to
thank you, quasibird;
does it really matter, if it is in- or out-side?
> If a triangle with integer side lengths a,b,c is such that,
> for some interior point P of the triangle, the 3 Cevians
> through P partition the triangle into 6 smaller triangles
> without all integer side lengths, then gcd(a,b,c) <= 1.
does it really matter, if it is in- or out-side?
> If a triangle with integer side lengths a,b,c is such that,
> for some interior point P of the triangle, the 3 Cevians
> through P partition the triangle into 6 smaller triangles
quasi
May 25, 2012, 8:00:13 PM5/25/12
to
1treePetrifiedForestLane wrote:
>quasi wrote:
>>
>>Conjecture:
I certainly don't want to let P be on the boundary of the
big triangle, otherwise the 6 small triangles would not all
be non-degenerate.
quasi
>quasi wrote:
>>
>>Conjecture:
>>
>> If a triangle with integer side lengths a,b,c is such that,
>> for some interior point P of the triangle, the 3 Cevians
>> through P partition the triangle into 6 smaller triangles
>> with all integer side lengths, then gcd(a,b,c) > 1.
>> If a triangle with integer side lengths a,b,c is such that,
>> for some interior point P of the triangle, the 3 Cevians
>> through P partition the triangle into 6 smaller triangles
>
>thank you, quasibird;
>does it really matter, if it is in- or out-side?
Maybe not, but I'm not sure.
>thank you, quasibird;
>does it really matter, if it is in- or out-side?
I certainly don't want to let P be on the boundary of the
big triangle, otherwise the 6 small triangles would not all
be non-degenerate.
quasi
quasi
May 29, 2012, 3:56:56 AM5/29/12
to
1000 satisfying the specified conditions.
It has sides
BC, CA, AB = 195, 244, 329
and Cevians AD, BE, CF meeting at point P such that
BD, CE, AF = 39, 183, 188
AC, BE, CF = 304, 282, 144
AP, BP, CP = 190, 141, 126
thus yielding a counterexample to the stated conjecture.
quasi
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