Product of inradius and circumradius?

Geometry Level 4

We are given a triangle with a perimeter of 272 and the product of its sides equal to 314. If the product of its inradius and circumradius can be written as $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

If you think no such triangle exists, type 1337 as your answer.

The answer is 429.

1 solution

Abc Xyz
May 15, 2016

Area of Triangle can be written in 2 ways :

1:Inradius*Semiperimeter

2:Product of sides/4(circumradius)

We can equate them both and solve

$\frac{314}{4*R}$ = $\frac{212}{2}$ *I

Therefore by solving the given relation we get

R*I= $\frac{314}{4*136}$ = $\frac{157}{272}$

So we have answer in $\frac{a}{b}$ form

Therefore a + b = 157 + 272 = $\boxed{429}$

Here, I show the existence of at least one such triangle. For the answer, refer to ABC Xyz's post.

Note that a triangle of sides $136-\epsilon , 136-\epsilon , 2\epsilon$ , where $\epsilon$ is a small positive number (say, $\epsilon < 0.5$ ) exists, and has perimeter $272$ . When $\epsilon$ tends to zero, the product of the sides also tends to zero. Also, at $\epsilon = 0.5$ , the triangle has product of sides $18360.25$ , which is large. SInce $2x(136-x)^2$ is continuous, therefore by the Intermediate Value Theorem(or so), for some $x$ in $(0,1)$ , the product of the sides will be $314$ , as desired.

Shourya Pandey - 5 years ago

Well explained bro..! +1.

Rishabh Tiwari - 5 years ago

Nice proof !!!

abc xyz - 5 years ago

Congrats brother, your solution is in the top solutions list :)

Ashish Menon - 5 years ago

@Ashish Menon – Thanks a lot !!! This is my first time in top solutions and I am going CRAZY due to maxima of happiness!!!!!!!!

abc xyz - 5 years ago

@Abc Xyz – Haha, nice comment

Ashish Menon - 5 years ago

How can you say that such a triangle exists?

It would be better if you could add a proof :)

Mehul Arora - 5 years ago

Sorry I am not THAT Smart !!! I don't know how to :(

abc xyz - 5 years ago

Solved the same way. Probably there is no other method !!

Niranjan Khanderia - 5 years ago

Nice (+1) @Sharky Kesa nice question.

Ashish Menon - 5 years, 1 month ago

Product of inradius and circumradius?

The answer is 429.

1 solution

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