Math Puzzle Column On Science 2.0
    By Sascha Vongehr | March 3rd 2013 09:23 PM | 8 comments | Print | E-mail | Track Comments
    About Sascha

    Dr. Sascha Vongehr [风洒沙] studied phil/math/chem/phys in Germany, obtained a BSc in theoretical physics (electro-mag) & MSc (stringtheory)...

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    Richard Mankiewicz, our man in Bangkok, also known as Red Man (see his profile – no no, not because of Bangkok’s red light district - that would be Stickman, not Red Man!) has started a Math Puzzle Column on Science2.0, first entry: Circles Stuck in a Triangle.

    Richard has another webpage where he proved to be able to come up with a lot more puzzles, so if we encourage him, he may well do a darn good job here, too - so go there and encourage him.

    He asks … , well go over there and have a look already.  Here is my solution, but I only tell you hints about how to get it:

    Consider that the circles touch the base of the triangle, thus the heights a, b, and c (he demands a > b > c) all go straight up (right angles) from the base of the outer triangle to the circle’s centers.  Circles are everywhere tangential to their radius, therefore, when two circles touch, the radius of one circle, say a, is on the same line as that of the touching one, say b, so they make a longer line, here a + b.  This gives a bunch of right triangles to play with (Next picture provided by Derek Potter; below I used Mathematica5 for the images):

    The Pythagoras theorem applied to the red triangle is

    (a + b)2 = (a - b)2 + [x + y]2

    where x and y are lengths (both parallel to the base of the triangle whose area is asked) of another two such triangles (blue and green in Derek's picture), thus resulting in

    (a + b)2 = (a - b)2 + [2 (b*c)0.5 + 2 (a*c)0.5 ]2

    Red Man wants integer numbers, but there are square roots(!).  Thus, the integers should be already squares.  The first case would be c = 1 (= 1*1), which leaves b = 4 (= 2*2), which makes also a = b*c = 4, and a triangle cannot be had (or it would have infinite area):



    The second case is c = 4, so b = 9, which makes a = b*c = 36:



    But Richard explicitly demands the next case, namely c = 9.  This leaves b = 16, and thus a = b*c = 144:



    Now again, an outer triangle cannot be had (if it's base is to still touch all three circles), or in other words, the “triangle’s” area would be infinite, because it opens up instead of having a top.

    Don’t agree?  Well maybe I am just trolling.  Always think for yourself!  Look at the links below for more Didactic Challenges.


    Update: Red Man does in fact not quite agree, because that all circles touch the base of the triangle was not really a strict requirement, and so he would not have to be red-faced now (which is of course no longer an option for Red Man):


    Actually you were on the right track, but your answer gave the infinite area triangle.
    I am an Architect & so barely numerate and simply cheated using my CAD software to go straight to a geometric solution (far easier than thinking). Simply adding 180 degrees to your triangle side walls however gave correct result: b = 20, and a =81
    Would attach a diagram, but this not allowed for plebs - but if you plot your numbers you will see for yourself.

    b = 20 results in a = 83.05, not 81.  Please send us a list of all the buildings you designed. ;-)
    You may well be right - I can offer no proof to the contrary - though if so you have identified a reasonably serious rounding error or other artefact with my CAD system's handling of geometry - certainly visual inspection at quite close range is consistent with those numbers within the resolution of the software.
    We architects do tend to validate things based on appearances though. :)

    Could be a coincidental *very* near miss that happens to give three whole number radii.
    Maybe time to reacquaint myself with high school maths...

    I just put your numbers in; only takes a minute.  Blowing up to occupy the whole screen, 83.05 fits, 81 leaves a visible gap.
    Hi Sascha

    I've just seen this and, firstly, thanks for the encouragement! You've had a good trip round the links so, yes, after teaching for 2 years I came down with a rare illness and have been at home for a year now - luckily the brain-deadening meds have stopped so have been able to think again the last 3 months or so. But obviously not that well...

    Yes, you're absolutely right, the restriction on whole numbers gives solutions that are easy to calculate numerically... but the geomtry changes!!

    One can still construct an isosceles triangle but upside-down, with the horizontal side touching the top of the large circle and the two sides tangent to the two larger circles. Indeed, the smallest circle is not involved in the triangle construction, but is needed to calculate the other two radii.

    Perhaps I should be "red faced" but, actually, if I put part(i) with c=4 and part (ii) with c=9, together as one question it makes it a better overall question. Many thanks!
    I suspected you mistyped "9" (the value of b) instead of "4", but decided to go with it because it allows to give the area without any formula, namely infinity, ha ha.
    What do you mean, "at home for a year now".  Home is still BKK, right?
    Yes, am still in BKK but my illness has meant I had to stop work - hence stuck at home in our condo. I'm actually much better now but not enough to risk a full-time job - homeostatic dysregulation is a secondary but potentially lethal symptom. And so have started to work on the net again. I actually enjoy making these problems and feels good to have my brain working again. I just hope I can start to see some income in a few months! I think it was Stephen Wolfram who quotes one of his professors as saying: "You can't make any money out of mathematics!"

    Are you still in China? Seemd like you were once looking for an exit :-)
    Still here in what is for now quite a paradise for many reasons.  Problem is: getting old and sick in China means, for example, not getting proper pain medication.  They simply do not give it to you, not even tramadol (which one can buy freely in BKK), no matter in how much pain you are - it is utterly ridiculous.  This situation is unlikely to improve, because in China's hospitals, the big problem is not the quality of your treatment, but that there are another 50 people in line behind you.