yes ...all the angles of the triangle must touch a spot on the circle..
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Yes, but which centre: the centroid, incentre or orthocentre? You need the incentre.
If the sides of the triangle are equal in length to the radius of the circle, then you can simply place the two ends of the hypotenuse on the perimeter of the circle, and it's remaining corner will lie at the circle's center.
the perpendicular bisector
You will need... A set of compasses (for drawing the circle), and a straight edge. First, draw the circle. Then - put the point of the the compasses on the line. Mark the circle where the pencil crosses it. Place the point on this intersection and repeat the action until you're back at your starting point. Now - using the straight edge, join every alternate point to each other - forming an equilateral triangle.
It can be but need not be.