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In-radius and altitudes of a scalene triangle

I submitted this problem to Brilliant but it got rejected so I decided to share it here. Anyone willing to solve the problem is welcome. Enjoy!

I am sorry but there was a mistake in the problem. Thanks to Gabriel W. for pointing it out. I had a different approach but after getting the answers I did not verify them by triangle inequality.

A scalene triangle has an in-radius of 1 cm. Given that the altitudes are positive integers when measured in centimeters, what is the only possible value of the sum of the altitudes?

Note by Sreejato Bhattacharya
4 years, 5 months ago

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The area of the triangle is (a+b+c)/2.

altitudes are integer:

(a+b+c) = ka = lb = mc, where k,l,m are integers

assume a >= b >= c

then k is either 3 (if they are all equal) or 2.

The equilateral case is easy to check: an equilateral triangle with inradius 1 has three altitudes of length 3.

When k = 2, a+b+c = 2a; b+c = a, which is impossible by triangle inequality.

Gabriel Wong - 4 years, 5 months ago

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Note that the lengths of the altitudes are integers, but the lengths of the sides of the triangle need not be integers. So, assuming that your reasoning is correct, I think \( k \) can take any real value between \( 2 \) and \( 3 \).

Sreejato Bhattacharya - 4 years, 5 months ago

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http://mathworld.wolfram.com/Inradius.html

The area is indeed (a+b+c)/2

(or just divide the triangle into three chunks ABI BCI CAI; its clear, given inradius is 1)

Gabriel Wong - 4 years, 5 months ago

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@Gabriel Wong Yes now I get it. Sorry for my mistake in the previous comment, I have edited it out. But note that the lengths of the sides of the triangle need not be integers themselves, so \( k \) can be any real number between \( 2 \) and \( 3 \).

Sreejato Bhattacharya - 4 years, 5 months ago

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@Sreejato Bhattacharya (a+b+c)/2 = 1/2 * a * altitude from A.

(a+b+c) = a*altitude from A (which is integer)

a+b+c = ka for integer a

Gabriel Wong - 4 years, 5 months ago

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