I ABC triangle with integer sides

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In the discussion about the ABC triangle with integer sides, participants explore the relationship between the triangle's incircle and the lengths of its sides. They calculate the radius of the incircle using the cosine theorem and properties of tangential quadrilaterals. The focus is on finding the integer values for the length of side BC, derived from a linear equation involving m and n, which represent segments related to the triangle's angles. Factorization of a derived equation reveals multiple integer solutions for m and n, leading to possible values for BC. The conversation emphasizes the importance of these geometric relationships in determining the integer side lengths of the triangle.
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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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