Theory
Definition. A circle that passes through all three vertices of a triangle is called the circumcircle of that triangle.
Theorem. The perpendicular bisectors of the sides of a triangle intersect at one point. This point is the center of the circumcircle of the triangle.
Problem set (reference)
Class problems as we used them on this topic. For interactive practice with solutions and progress tracking, use Problems.cc when available.
1. How many circumcircles can be drawn around a triangle?
2.
a) Where is the center of the circumcircle located in a right-angled triangle?
b) The center of the circumcircle lies inside the triangle. Prove that this triangle is acute-angled.
3. Two chords of a circle form equal angles with its diameter. Prove that these chords are equal. Is the converse true?
4. From the midpoint of the hypotenuse, a perpendicular is drawn to intersect the leg. The point of intersection is connected to the end of the other leg by a segment that divides the angle of the triangle in a ratio of 2:5 (the smaller part is adjacent to the hypotenuse). Find this angle.
5. Given a segment AB. Find the geometric locus of points C such that triangle ABC is right-angled.
6. On the sides of angle ∠ABC, which equals 120°, segments AB = BC = 4 are laid off. A circle is drawn through points A, B, and C. Find its radius.