This conjecture follows from a discussion about the construction of an isoceles triangle given only a base.
The point that is placed on the base of this isoceles triangle is the midpoint. Notice that the point that creates the vertex of the triangle seems to be on the same line. In addition, the line that contains the midpoint and the "vertex" point is perpendicular to the base.
Click here to see the construction of an isoceles triangle.
This information can be used to construct the isoceles triangle for our tangent circle.