Question

In a triangle ABC, \(h_1,~h_2,~h_3\) are the lengths of altitudes from A, B and C, which are natural numbers forming an increasing H.P. If \(h_2\) = 24, then possible value(s) of a:b:c is/are ___.

Answer 1

angles?

Answer 2

Nothing given.

Answer 3

no no
a:b:c

Answer 4

what do u mean by H.P ?

Answer 5

But, this is given that all the altitudes are natural numbers.

@aajugdar, no they are for sides.

@BSwan , it means harmonic progression.

Answer 6

harmonic progression

Answer 7

ive seen somethinglike this before mmm
but that was dealing with vectors
let me think if its gonna well with this

Answer 8

HP = inverse of arithmetic series

Answer 9

That is \(\large{\cfrac{1}{h_1} + \cfrac{1}{h_3} = \cfrac{2}{h_2}}\)

Answer 10

Since, \(h_2 = 24\)

\(\implies \large{\cfrac{1}{h_1} + \cfrac{1}{h_3} = \cfrac{1}{12}}\)

Answer 11

use sine law :D

Answer 12

Okay

Answer 13

you have a diophantine equation now, with \(h_1<h_3\) am i right?

Answer 14

So, by sine law I have,

\(\large{\cfrac{\sin{C}}{h_1} = \cfrac{1}{b}}\)

\(\implies \large{\cfrac{1}{h_1} = \cfrac{1}{b*h_1}}\)

@mukushla, well it is not in the syllabus of the exam I was preparing for. So, we cannot use it.

Answer 15

you just need to consider restrictions on \(h_1\) and \(h_3\)

Answer 16

@mukushla can we solve equation of type `ax + by + cxy = d` using diophantine method ?

Answer 17

Sorry guys I have to go. I will post my side of try when I am back.

Until then, goo day.

Answer 18

ok, let us not call it diophantine :-) it's just an equation with natural variables. @vishweshshrimali5 see you later ;) @ganeshie8 oh yeah, do you have the general olution for that type of diophantine?

Answer 19

why do u say
1/h1 + 1/h3 = 2/h2
isnt HP in the form
1/h1 + 1/(h1+d) + 1/(h1+2d)=1/h1 + 1/h2 + 1/ h3

Answer 20

Wow! thats interesting xD I am familiar with solving equations like ax+by =c using diophantine... but never dealt with an equation involving second order terms...
I'll open a new thread :)