Circles in a triangle
Math brain teasers require computations to solve.
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(sa)(sb)(sc)). This gives us x = sqrt((sa)(sb)(sc)/s).
The radius of the circumscribed circle is given by R = abc/4A.
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Comments
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 = âˆšs(sa)(sb)(sc). 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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