The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.
Problem
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Tags: geometry, area, perimeter, circumcircle, inradius
07.02.2019 23:25
Let the perimeter be n, side of the triangle are a,b,c. Then area =$\frac{abc}{4R}=rs =>\frac{4abc}{65}=n$ Thus 13 divides exactly one of a,b,c and same for 5. From here, hopefully we can solve? (Sorry for the incomplete solution)
12.02.2019 18:31
adik7 wrote:
. I'll add in an actual solution later. I’m sure you will!
12.02.2019 20:28
You also need to prove that $\lceil \min(p)\rceil=\lfloor \max(p)\rfloor$
12.02.2019 23:04
$a,b,c<2R<17$ $abc=4SR=2PRr=65P$ Let $13|a$ obvious, that $a=13$ $bc=5(13+b+c)$ Let $b=5d,d<4$ $cd=13+5d+c$ $(c-5)(d-1)=18$ If $d=2,c=23$ but $c<17$ If $d=3,b=15,c=14$ Answer$(13,14,15)$
13.02.2019 00:21
unless there is some typo we have in the statement that the perimeter is an integer but it s not mentioned that $a,b,c$ are natural numbers