### Brain Teasers

# Circles in a triangle

Find the radius of the inscribed and circumscribed circles for a triangle.

### Answer

Let a, b, and c be the sides of the triangle. Let s be the semiperimeter, i.e. s = (a + b + c) / 2. Let A be the area of the triangle, and let x be the radius of the incircle.Divide the triangle into three smaller triangles by drawing a line segment from each vertex to the incenter. The areas of the smaller triangles are ax/2, bx/2, and cx/2. Thus, A = ax/2 + bx/2 + cx/2, or A = sx.

We use Heron`s formula, which is A = sqrt(s(s-a)(s-b)(s-c)). This gives us x = sqrt((s-a)(s-b)(s-c)/s).

The radius of the circumscribed circle is given by R = abc/4A.

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## Comments

ok then...

dude, A teaser would have been nice.

Here we have a good example of the inadequacy of the present rating system for the teasers, which is by users choosing one of three levels.

This one should be rated: "for those who have mastered at least High School geometry".

The explanation of the answer is incomplete, and ought to have included a link to a relevant online geometry textbook.

This one should be rated: "for those who have mastered at least High School geometry".

The explanation of the answer is incomplete, and ought to have included a link to a relevant online geometry textbook.

Boy was Braingle ever desperate in their first couple weeks of being open to teasers.

Shouldn't it be Hero of Alexandria's Formula, not Heron?

yeah........

that teaser scared me!!

all of the above!!!

hate to be a jerk, it says find the radius of the circle, right? well, it doesn't give us numbers, so we want it in variables right? how about "x", teaser done with. Reread the text to see what i mean. the answer is X

Yikes I'm a freshman in college as a math major and did't know how to solve that (never heard of the formula you used either) I think I'll get back to my studies.....

Needs a better description of terminology. For example, explanation of misc. variables. For the sake of brevity, it is obvious that some idiosyncracies were ignored by the author. Good intentions with it though!

Great teaser .....I think, but good one....brain now fried

what ?

Really confusing.....still working on it....doesn't make much sense to me...

I liked that problem. I must admit that I coudn't get the answer myself, but the explanation did make sense to me, a high school sophomore. I have only ever heard of Heron of Alexandria, not Hero. And a college student, let alone a math major, who is not familiar with Heron's formula is truly saddening.

Heron's formula and Hero's formula are the same thing. A = âˆšs(s-a)(s-b)(s-c). It's not commonly taught for some reason. I had to stumble upon it when I went through every math book I could find in my college library. This teaser is hard, but the conclusion is correct.

hey man your brain is too big!!! i thought that the answer will be a constant number... hahaha guess need to study more....

i thought you could only find the radius and circumfrence of a circle

If I want this sort of question I'll open my geometry text book. A worthwhile question in context but is it really a suitable Braingle Teaser? Oh well, I enjoyed working it out!

The answer to this teaser should be "Make me."

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