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?

#Geometry #NumberTheory #Logic #Science #MathProblem #Math #Opinions

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 8 years, 1 month ago

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 - 8 years, 1 month ago

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 - 8 years, 1 month ago

@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 - 8 years, 1 month ago

@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 - 8 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$