I've attempted various solutions for the problem below, to no avail:

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle?

Methods attempted:

  1. First, I tried drawing the problem "bare-bones". I tried finding the area of the segment using: $\frac { \theta }{ 360 } *\pi { r }^{ 2 }-[\triangle BOC]\quad =\quad area\quad of\quad segment$. I would divide this by the area of an equilateral triangle ($\frac { { s }^{ 2 }\sqrt { 3 } }{ 4 } $). enter image description here
  2. I then tried extending the tangent lines to form another equilateral triangle, as such: enter image description here
  3. Lastly, (very similar to previous), I tried making similar triangles ($\triangle ADE\sim \triangle ABC$), as shown below, but quickly realized it wouldn't help. enter image description here

I take your second figure as reference, for the naming of points. But this naming doesn't coincide with the naming in the statement of the initial question (where $B$ and $C$ are for the points of tangency).

Let $a$ be the side of the equailateral triangle, and $r$ the radius of the circle, with:


The area of equilateral triangle ADE is


The value of the area of triangle ADE minus the circular segment is

$$\tag{2}\underbrace{ar}_{\text{area of quadrilateral ODEA}}-\underbrace{(2 \pi/3)r^2}_{\text{area of circ. sector with angle 120°}}$$

Note the use of radians, without reference to degrees.

It suffices now to take the ratio between (2) and (1), and take (0) into account, yielding a unit-free result (a constant, independent of $r$ or $a$).

  • $\begingroup$ Thanks for the clear explanation, but the problem only asks for the area of triangle ADE outside of the circle, so why do we add r^3sqrt(3)/4, for (2)? $\endgroup$ – DarkRunner Oct 12 '17 at 0:29
  • $\begingroup$ You are right. I don't know where I had my head ... I correct it. $\endgroup$ – Jean Marie Oct 12 '17 at 0:35
  • $\begingroup$ Thanks, and by the way, is there a name for the triangle within a triangle, such as triangle DEF within ABC? $\endgroup$ – DarkRunner Oct 12 '17 at 0:40
  • $\begingroup$ In this special case where the inscribed circle is tangent to the sides in their midpoints, it is called the medial triangle or midpoint triangle (en.wikipedia.org/wiki/Medial_triangle) $\endgroup$ – Jean Marie Oct 12 '17 at 7:17

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