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Circles in a triangle

Math brain teasers require computations to solve.


Puzzle ID:#142
Fun:** (1.67)
Difficulty:**** (3.17)
Submitted By:offsky**
Corrected By:solarsistim321




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

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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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Dec 06, 2002

ok then...
Nov 19, 2003

dude, A teaser would have been nice.
Jul 13, 2004

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.
Feb 11, 2005

Boy was Braingle ever desperate in their first couple weeks of being open to teasers.
May 11, 2005

Jun 08, 2005

Shouldn't it be Hero of Alexandria's Formula, not Heron?
Nov 09, 2005

Jan 25, 2006

that teaser scared me!!
Jan 26, 2006

all of the above!!!
Jan 29, 2006

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
Feb 20, 2006

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.....
Mar 13, 2006

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!
Mar 18, 2006

Great teaser .....I think, but good one....brain now fried
Apr 05, 2006

what ?
Apr 12, 2006

Really confusing.....still working on it....doesn't make much sense to me...
May 11, 2006

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.
Jun 14, 2006

Heron's formula and Hero's formula are the same thing. A = √s(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.
May 22, 2007

hey man your brain is too big!!! i thought that the answer will be a constant number... hahaha guess need to study more....
Jun 28, 2008

i thought you could only find the radius and circumfrence of a circle
Oct 11, 2009

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!
Sep 07, 2013

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

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