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## Circles in a triangle

Math brain teasers require computations to solve.

 Puzzle ID: #142 Fun: (1.68) Difficulty: (3.16) Category: Math Submitted By: offsky Corrected By: solarsistim321

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

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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 speedyg1000 Dec 06, 2002 ok then... Mogmatt16 Nov 19, 2003 dude, A teaser would have been nice. canu 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. Sane Feb 11, 2005 Boy was Braingle ever desperate in their first couple weeks of being open to teasers. darthforman May 11, 2005 brianz Jun 08, 2005 Shouldn't it be Hero of Alexandria's Formula, not Heron? nkatha23 Nov 09, 2005 yeah........ SPUTNIK2 Jan 25, 2006 that teaser scared me!! chata Jan 26, 2006 all of the above!!! lessthanjake789 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 keveffect1 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..... say2joe 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! redraptor50 Mar 18, 2006 Great teaser .....I think, but good one....brain now fried Vigo95 Apr 05, 2006 what ? mmmcla01 Apr 12, 2006 Really confusing.....still working on it....doesn't make much sense to me... Atypic4l 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. udoboy 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. jinzcarmela 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.... nascarfan19 Jun 28, 2008 i thought you could only find the radius and circumfrence of a circle dalfamnest 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! eighsse Sep 07, 2013 The answer to this teaser should be "Make me."

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