Question

a, b, c, d - positive numbers. Prove that there is a triangle that has sides of such lengths:
square root of (b^2 + c^2), square root of (a^2 + c^2 + d^2 + 2cd), square root of (a^2 + b^2 + d^2 + 2ab) and find the area of this triangle.

Answer 1

\[\sqrt{b^2 + c^2}, \sqrt{a^2 + c^2 + d^2 + 2cd}, \sqrt{a^2 + b^2 + d^2 + 2ab}\]

Answer 2

I notice that the second length can be simplified to: \(\sqrt{a^2 + (c +d)^2}\) and the third length to: \(\sqrt{d^2 + (a + b)^2}\)

Answer 3

However, I have no idea what to do with this knowledge.

Answer 4

What's the essential thing for a triangle to be made?

Answer 5

Perhaps a base? Or the height? (Not sure how it is called in English, can't find the translation, the thing that is usually denoted with a letter h)

Answer 6

Hypotenuse!

Answer 7

But what do I get from that knowledge?

Answer 8

Lol! No , i am asking for a particular property that guides the formation of a triangle.

Answer 9

You must be familiar with triangle inequality.

Answer 10

Well, I think that you are talking about the idea that in atriangle there are two or one triangles with one angle of 90 degrees? So it is possible to apply Pythagorean theorem?
I am familiar with it, but how would it help there?

Answer 11

Okay! Let's start with basics .

Answer 12

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

Answer 13

Have a look at it.

Answer 14

So, first I would take any of the lengths, make up such inequality, square both sides and look what goes there? But since I don't know the numeric values of letters, how would I know, if the inequality holds? (I need to prove that those lengths really make up a triangle)

Answer 15

We don't need that rather . There will be a lot much simplification after you do that process .

Answer 16

I will try to do that:
\(\sqrt{a^2 + b^2} < \sqrt{(c+d)^2 + a^2 }+\sqrt{(a+b)^2 + d^2}\)
Squaring both sides:
\(a^2 + b^2 < (c +d)^2 + a^2 + 2(\sqrt{(c+d)^2 + a^2 })(\sqrt{(a+b)^2 + d^2}) + (a +b)^2 +d^2\)

Answer 17

So:
\(0 < (c+d)^2 + 2(\sqrt{(c+d)^2 + a^2 })(\sqrt{(a+b)^2 + d^2}) + a^2 + 2ab +d^2\)

Answer 18

I think that you did a mistake .

Answer 19

Where?

Answer 20

It was square root of b^2+c^2 and not of a^2+b^2.

Answer 21

Ahh! My bad. Got ahead of myself a little bit with those letters ;)

Answer 22

Happens!

Answer 23

But exchanging last terms of RHS a to c, and we got proof? Or do I need to do that for all sides?

Answer 24

The first half would do ..

Answer 25

Sorry! It's getting l8 here.. I am really sleepy at the moment.

Answer 26

No problem! You can go sleep, just hint me how I would get the area ;)

Answer 27

Heron's formula

Answer 28

Thank you very much!

Answer 29

Welcum...:)