True, and it's called an isosceles right triangle :D
D) construction of the angle bisector.
So if you have a triangle that is not equilateral the perpendicular bisectors would not work so you need angle bisectors.
I have drawn a diagram
The green is the angle bisectors and the yellow the perpendicular bisectors.
You can see the problem straight away
Answer: The answer is converges r=1/7
Warning: This might be rough...
First draw it out. Label the angles at the corners of the triangle 60 (definition of equilateral triangles). Now draw a line from the center of the circle to the corner, splitting the corner in half. Label this line R and a corner as 30 degrees. No to find the height of this triangle, you do rsin(30). The base of this triangle is 2rcos(30). Now find the area of this mini triangle (rsin(30)*2rcos(30)/2=r/2*rsqrt(3)/2=r^2sqrt(3)/4). Now multiply this by 3 because you have 3 mini triangles... to get...
<span>r^2 3sqrt(3)/4</span>