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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.
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.
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