I ABC triangle with integer sides

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TL;DR Summary
In the ABC triangle with integer sides, point D on [AB] and point E on [AC] are taken such that DE is tangent to the incircle. If ∣AD∣=5, ∣AE∣=6, ∣DE∣=7, how many different values can ∣BC∣ take?
I can find radius of incircle of ADE and incircle of ABC triangle but I don't know how them help me.
 
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littlemathquark said:
TL;DR Summary: In the ABC triangle with integer sides, point D on [AB] and point E on [AC] are taken such that DE is tangent to the incircle. If ∣AD∣=5, ∣AE∣=6, ∣DE∣=7, how many different values can ∣BC∣ take?

I can find radius of incircle of ADE and incircle of ABC triangle but I don't know how them help me.
We know cos A by cosine theorem. Does it help you ?
 
anuttarasammyak said:
We know cos A by cosine theorem. Does it help you ?
Cos A=1/5 but I can't use it.
 
If BC length would not integer, perhaps there would be infinite value of BC I think. But it's integer.
 
How much is radius of incircle of triangle ABC you get ? I am interested in how you get it.
 
anuttarasammyak said:
How much is radius of incircle of triangle ABC you get ? I am interested in how you get it.
Let AKL be isosceles triangle with AK=AL=9
We easly find KL using cosine theorem in AKL triangle. (cosA=1/5)
Let O be incenter of ABC triangle and R ve radius. Cos O=-cosA=-1/5
Using cosine theorem in OKL we can find R but I have'nt got paper and pencil now, I'm out.
 
littlemathquark said:
Let AKL be isosceles triangle with AK=AL=9
Thanks. From where 9 comes ?
 
Using tangent properties
DE=4+3=7
AK=5+4=9
AL=6+3=9
 
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You also get a linear relation for the sides of the triangle from the fact that the circle is inscribed the the BCED quadrilateral.
 
  • #10
martinbn said:
You also get a linear relation for the sides of the triangle from the fact that the circle is inscribed the the BCED quadrilateral.
İt's tangential quadrilateral.
 
  • #11
littlemathquark said:
Using tangent properties
DE=4+3=7
AK=5+4=9
AL=6+3=9
I see. Thanks. So
R=9 \tan \frac{A}{2}=\frac{9\sqrt{2}}{\sqrt{3}}
Is it same as yours ?

m=R cot \frac{B}{2}, n=R cot\frac{C}{2}
Cosine theorem
(9+m)^2+(9+n)^2-2(9+m)(9+n)cos A=(m+n)^2
 
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  • #12
littlemathquark said:
İt's tangential quadrilateral.
Yes. That's what I am saying. But the point is that this gives you a linear equation and the cosine theorem gives you a quadratic equation. The system probably doesn't have many integral solutions.
 
  • #13
anuttarasammyak said:
I see. Thanks. So
R=9 \tan \frac{A}{2}=\frac{9\sqrt{2}}{\sqrt{3}}
Is it same as yours ?

m=R cot \frac{B}{2}, n=R cot\frac{C}{2}
Cosine theorem
(9+m)^2+(9+n)^2-2(9+m)(9+n)cos A=(m+n)^2
Yes R is the same as mine. By simplified your cosine theorem I found (if I'm not mistake) m+n=mn/6 - 9
 
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  • #14
littlemathquark said:
Yes R is the same as mine. By simplified your cosine theorem I found (if I'm not mistake) m+n=mn/6 - 9
From this equation (m-6)(n-6)=90 and it can be found m, n integer and values of m+n. Is it ok?
 
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  • #15
Factorization of 90 will tell you how many cases.
 
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  • #16
anuttarasammyak said:
Factorization of 90 will tell you how many cases.
Suitible ##(m,n)## is ##(7,96),(8,51),(9,36),(11,24),(12,21),(15,16)## and ##m+n=|BC|=\{103,59,45,35,33,31\}##
 
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  • #17
(m-6)(n-6)=2*3^2*5

so m-6 ={1,2,3,5,...} : 2*3*2=12 ways
 
  • #18
Thank you, but m,n is symmetric and we're interested in m+n
 
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