The perimeter of a triangle is a natural number, its circumradius is equal to 658, 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 =abc4R=rs=>4abc65=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 ⌈min
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