Theory
Criteria for a Quadrilateral to be Inscribed:
- A circle can be inscribed in a convex quadrilateral if and only if the angle bisectors of three of its angles intersect at one point.
- A circle can be inscribed in a convex quadrilateral if and only if the sums of the lengths of its opposite sides are equal.
Problems
- Determine the type of parallelogram into which a circle can be inscribed.
- A circle is inscribed in an isosceles trapezoid. Prove that the lateral side of the trapezoid is equal to its midline.
- Given a trapezoid into which a circle can be inscribed. Prove that circles constructed on the lateral sides as diameters touch each other.
Definition: A quadrilateral with two pairs of equal adjacent sides is called a kite (or deltoid).
- a) Can a circle always be inscribed in a kite?
b) Prove that one of the diagonals of a kite is its axis of symmetry. - Prove that if there exists a circle that touches all sides of a convex quadrilateral ABCD, and another circle that touches the extensions of all its sides, then the diagonals of such a quadrilateral are perpendicular.