3
$\begingroup$

Please help me show that: the equilateral triangle of altitute $3r$ is the isosceles triangle of least area circumscribing a circle of radius $r$.

Iassumed the following:

base = $2a$

height = $h$

radius of circle = $r$

Area = $\frac{1}{2}(2a)h = ah$

$tan(2\theta) = \frac{h}{a}$ and $tan\theta = \frac{r}{a} $

Ii tried using double angle identities to represent h in terms of a and \theta but I have failed miserably.

This is what I ended up with: $\displaystyle{% h={2r\cos^{2}\left(\theta\right) \over 2\cos^{2}\left(\theta\right) - 1}}$

I know i need to differentiate the area and set to zero, but I can't get it. please direct me how to solve this problem. Thank you;

$\endgroup$
  • $\begingroup$ I think you do not yet have the correct expression, as the minimum is not at the correct value of $ \ \theta \ $ . The expression I find for the area is rather more complicated; there is a minimum at the proper value, but solving the derivative function for its zeroes is unpleasant. I will post once I have a decent solution... $\endgroup$ – colormegone Jun 25 '14 at 22:07
2
$\begingroup$

This is a nice example of a problem where the "right" choice for the independent variable makes a solution reasonably tractable, whereas other choices will lead to rather rough going. I tried a straight-ahead Cartesian approach (which led to a quintic equation) and the trigonometric approach you attempted (which gave a derivative function running up to fourth powers of sine which was going to be difficult to solve -- though it could be shown that one would obtain the expected angle for an equilateral triangle).

enter image description here

Instead we will work with the slope $ \ m \ $ of the sides of the isosceles triangle. If we center the circle of radius $ \ r \ $ on the origin, the "base" will lie along the line $ \ y \ = \ -r \ $ ; we will work with one vertex at $ \ ( \ a, \ -r \ ) \ $ , continuing your notation. The apex of this triangle is then located at $ \ ( \ 0, \ h \ - \ r \ ) \ $ and the area of the triangle is $ \ A \ = \ ah \ $ . The slope of the marked side is then $ \ m \ = \ -\frac{h}{a} \ $ .

We need some sort of constraint for this optimization: that is provided by the fact that the circumscribed triangle must be tangent to the circle on each of its sides. We will call the tangent point on the marked side $ \ ( \ X, \ Y \ ) \ $ . The equation of the line along which this side lies is $ \ y \ = \ ( \ h \ - \ r \ ) \ - \ \frac{h}{a} x \ $ .

Since the radius of the circle which extends to the tangent point is perpendicular to the tangent line, the equation of the line for this radius is $ \ y \ = \ -\frac{1}{m} x \ $ . Its intersection with the tangent line is thus given by

$$ \ -\frac{1}{m} X \ = \ ( \ h \ - \ r \ ) \ + \ m X \ \ \Rightarrow \ \ X \ = \ -\frac{m}{m^2 \ + \ 1} \ ( \ h \ - \ r \ ) $$ $$ \Rightarrow \ \ Y \ = \ \frac{1}{m^2 \ + \ 1} \ ( \ h \ - \ r \ ) \ \ . $$

Now we apply the constraint: the tangent point lies on the circle, so

$$ X^2 \ + \ Y^2 \ = \ r^2 \ \ \Rightarrow \ \ \left[ \ \left(-\frac{m}{m^2 \ + \ 1}\right)^2 \ + \ \left(\frac{1}{m^2 \ + \ 1}\right)^2 \ \right] \ ( \ h \ - \ r \ )^2 \ = \ r^2 $$

$$ \Rightarrow \ \ \left(\frac{1}{m^2 \ + \ 1}\right) \ ( \ h^2 \ - \ 2rh \ + \ r^2 ) \ = \ r^2 \ \ . $$

If, for the moment, we call $ \ \mu \ = \ \frac{1}{m^2 \ + \ 1} \ $ , then we can solve the quadratic equation $ \ \mu \ h^2 \ - \ 2 \mu r \ h \ + \ ( \ \mu \ - \ 1 \ ) r^2 \ = \ 0 \ $ to obtain

$$ h \ = \ \frac{2 \mu r \ \pm \ \sqrt{4 \mu^2 r^2 \ - \ 4 \mu \ ( \mu \ - \ 1 \ ) \ r^2}}{2 \mu} \ = \ r \ \left( 1 \ \pm \ \frac{1}{\sqrt{\mu}} \right) \ \ ; $$

since it is plainly the case that $ \ h \ > \ r \ $ , we discard the "negative solution" to produce $ \ h \ = \ r \ ( \ 1 \ + \ \sqrt{m^2 \ + \ 1} \ ) \ $ .

We have made our way through the first thicket and prepare to enter the second one. The area function is now found from

$$ m \ = \ -\frac{h}{a} \ \ \Rightarrow \ \ a \ = \ -\frac{h}{m} $$

$$ \Rightarrow \ \ A \ = \ ah \ = \ -\frac{h^2}{m} \ = \ - r^2 \ \left( \frac{m^2 \ + \ 2 \ + \ 2\sqrt{m^2 \ + \ 1}}{m} \right) \ \ . $$

It is this latter function for which we wish to locate the "critical value" for $ \ m \ $ ; hence,

$$ \frac{dA}{dm} \ = \ - r^2 \ \left( 1 \ - \ \frac{2}{m^2} \ - \ \frac{2}{m^2 \ \sqrt{m^2 \ + \ 1}} \right) $$

$$ = \ - r^2 \ \left( \frac{m^2 \ \sqrt{m^2 \ + \ 1} \ - \ 2 \sqrt{m^2 \ + \ 1} \ - \ 2} {m^2 \ \sqrt{m^2 \ + \ 1}} \right) \ = \ 0 \ \ . $$

Since the denominator cannot be zero (as $ \ m \ $ is non-zero), we only need to consider when the numerator is zero:

$$ m^2 \ \sqrt{m^2 \ + \ 1} \ - \ 2 \sqrt{m^2 \ + \ 1} \ - \ 2 \ = \ 0 \ \ \Rightarrow \ \ (m^2 \ - \ 2 ) \ \sqrt{m^2 \ + \ 1} \ = \ 2 $$

$$ \Rightarrow \ \ m^2 \ + \ 1 \ = \ \frac{4}{(m^2 \ - \ 2)^2} \ \ \Rightarrow \ \ m^6 \ - \ 3 \ m^4 \ = \ m^4 \ ( \ m^2 \ - \ 3 \ ) \ = \ 0 \ \ . $$

[squaring both sides of the equation and simplifying in this last line]

We have already rejected zero as a possible solution, so we have $ \ m \ = \ \pm \sqrt{3} \ $ . But these are the slopes of the sides of an equilateral triangle above the (horizontal) base; this then is our circumscribed isosceles triangle of minimal area.

$\endgroup$
  • $\begingroup$ Hi @RecklessReckoner, thanks for your solution and diagram. can you please explain how you got the following step: h = r(1 +/- (1/sqrt(u)) => h = r(1 + sqrt(m^2 + 1) ?? I see it as: h = r(1 + 1/(sqrt(m^2 +1)) $\endgroup$ – Spencer Trinh Jun 26 '14 at 18:30
  • $\begingroup$ Because we declared $ \ \mu \ = \ \frac{1}{m^2 \ + \ 1} \ $ , so $ \ \frac{1}{\sqrt{\mu}} \ = \ \sqrt{m^2 \ + \ 1} \ $ . $\endgroup$ – colormegone Jun 26 '14 at 21:59
0
$\begingroup$

First, two corrections in the question

1) The height will be $3r/2$

2) This will maximize the area. Since the minimum area of an isosceles triangle will be of height $2r$ and zero base, hence zero area.

Illustration of isosceles triangle circumscribed in circle of radius $r$

From the figure, $h=r+\sqrt{r^2-a^2}$, area $A=ah=a(r+\sqrt{r^2-a^2})$. Now equate $\frac{\partial A}{\partial a}=0$ to get the $a$ which maximizes the area. For your info, $$ \frac{\partial A}{\partial a}= \frac{r\sqrt{r^2-a^2} + r^2-2a^2}{\sqrt{r^2-a^2}}; \\ \frac{\partial A}{\partial a}=0 \Rightarrow r\sqrt{r^2-a^2} + r^2-2a^2 = 0 \\ (2a^2-r^2)^2 = r^2(r^2-a^2) \\ a = \frac{\sqrt{3}}{2}r $$

Hope this helps.

$\endgroup$
  • $\begingroup$ Hi Vijay, thank you very much for the beautiful diagram and your work. however, from what i understand from the wording, the isosceles triangle is crcumscribing the circle and not vice versa as you have shown. correct me if I am wrong. i tried googling it and it appears to be the triangle is outside of the circle. $\endgroup$ – Spencer Trinh Jun 25 '14 at 15:15
  • $\begingroup$ The problem for the in scribed triangle is pretty straightforward. Solving for the circum scribed triangle is not at all pretty (I've tried it two ways so far and am still looking for a nicer solution before I post anything). $\endgroup$ – colormegone Jun 25 '14 at 22:02
  • $\begingroup$ Interpreted wrongly :P $\endgroup$ – Vijay Jun 27 '14 at 11:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.