Inscribed and Circumscribed Triangles

A circle that circumscribes a triangle is a circle containing the triangle such that the vertices of the triangle are on the circle.

A circle that inscribes a triangle is a circle contained in the triangle that just touches the sides of the triangle.

Circumscribing a triangle.
Here is a method for constructing the circle that circumscribes a triangle.

  1. Draw the triangle.
  2. Draw the perpendicular bisector to each side of the triangle. Draw the lines long enough so that you see a point of intersection of all three lines.
  3. Draw the circle with radius at the intersection point of the bisectors that passes through one of the vertices. You should see that this circle passes through all three vertices, and that it is the desired circle.

Inscribing a triangle.
Here is a method for constructing the circle that inscribes a triangle.

  1. Draw the triangle.
  2. Draw the angle bisector for each angle of the triangle. Draw the lines long enough so that you see a point of intersection of all three lines.
  3. Draw a line perpendicular to any side that passes through the intersection point. Mark the point on the side through which this line passes.
  4. Draw the circle with radius at the intersection point that passes through the point you obtained in the last step. This is the desired circle.

Assignment: Draw two triangles of different shapes and then construct the circle that circumscribes them. Next, draw two triangles and then construct the circle that inscribes them.