# square inscribed in a right triangle

A square of maximum possible area is circumscribed by a right angle triangle ABC in such a way that one of its side just lies on the hypotenuse of the triangle.
What is the area of the square? actually the answer is given as $(abc/(a^2+b^2+ab))^2$ Please provide the approach to solve the problem.

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Presumably, $a$, $b$, and $c$ are the sides of the triangle, with $c$ being the hypotenuse? – Gerry Myerson Mar 4 at 12:11
What have you tried? – dtldarek Mar 4 at 12:11
Seems to me (draw a picture!) that there are several triangles similar to ABC, and you should be able to get some mileage from that. – Gerry Myerson Mar 4 at 12:13
@dtldarek i extended the triangle into a rectangle. and the square also. Then i used $1/2*d_1*d_2$ and the final answer as $ab/4$. – cdummy Mar 4 at 12:20
I don't understand how you extend the triangle and square, and I don't know what you mean by $d_1$ and $d_2$. Also, you haven't answered my question about the meanings of $a$, $b$, and $c$. – Gerry Myerson Mar 4 at 12:50
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Consider - a,b as right legs and c as the hypotenuse.

Let side of square = s AC = b, BC = a, AB = c.

FB = as/b and AE = bs/a as the colored triangles are similar to the bigger triangle.

Steps to calculate area (S^2) :

1)Calculate GB and AD using right angle triangle rule for triangles GBF and ADE.

2)Calculate GD using right angle triangle rule for triangle GCD.

3)GD^2 = s^2. You get a quadratic equation in s which can be factorized. You get s = (abc)/(a^2 + b^2 +a.b)

If still not clarified will post the answer then.

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 how you got FB=as/b – cdummy Mar 4 at 15:49 i got $FB=bs/a$ and $EA=sa/b$ and please post the clarification. – cdummy Mar 4 at 16:45 sry you are correct.. i just wrongly used the ratio. FB=as/b is correct.And thanks for the answer. I got it. – cdummy Mar 4 at 18:05

Hint:

$\hspace{60pt}$

The red solid line is the height dropped onto the hypotenuse, i.e. $h = \frac{ab}{c}$ and the red dotted lines are of the same length. The green parallel lines are unnecessary, but might get you some intuitions.

Good luck! ;-)

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how to get $h=ab/c$. which pair of similar triangles to consider for that. – cdummy Mar 4 at 15:39
@cdummy $P_{\triangle ABC} = \frac{1}{2}ab = \frac{1}{2}ch$. – dtldarek Mar 4 at 16:09
I appreciate your answer from my heart. But the problem is i need some simple solution. or a hard one that is explained. See your reputation is 6340, where as mine is 29. i m sry to say this i cant get your point of approach. Can u give a detailed solution only if u are interested... – cdummy Mar 4 at 16:30
@cdummy Well, I don't have time to write a fully detailed answer, but consider the following. First calculate the area of the triangle in two ways: $\frac{1}{2}ab = |\triangle ABC| = \frac{1}{2}ch$ to get $h = \frac{ab}{c}$. Let $x$ be the side of a square, then the picture implies (the black and gray triangles are similar) that $\frac{x}{c} = \frac{h}{c+h}$, so $$x = \frac{ch}{c+h} = \frac{c\frac{ab}{c}}{c+\frac{ab}{c}} = \frac{abc}{c^2+ab}.$$ – dtldarek Mar 4 at 17:03
thanks very much – cdummy Mar 4 at 17:09
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Hint: The square formed some similar right-angled triangles, make use of the ratio of the sides.

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 Much as I noted earlier, in one of my comments. – Gerry Myerson Mar 4 at 12:51