Analyze the provided image and extract the relevant information about geometry problems and figures. The image contains mathematical problems and diagrams related to circles and angles. The task is to identify and describe these elements according to standard geometric analysis.

The image displays problems and diagrams related to geometry, specifically focusing on circles, angles, and their properties. The problems are numbered 781, 782, and 783, and there are several figures labeled with letters a) through ж) accompanying problem 783.

Problem 781: This problem, in Russian, refers to a previous task (Task 70) and asks to construct a segment that is the geometric mean of two given segments, using a stated theorem.

Problem 782: This problem states that the sides of an angle touch two circles at point A, and they have a common tangent. It asks to prove that the centers of these circles lie on the line OA.

Problem 783: This problem presents circles and angles drawn on a grid. It asks to find the measure of angle ABC based on the data in Figure 267. There are seven figures labeled a) through ж) which are illustrations for this problem.

Figure Analysis for Problem 783:

• a) A circle with center O and points A, B, C. A line segment connects A to B, and another connects B to C. A radius OA is shown, and a line segment from O to B. Angle ABC appears to be inscribed in the circle.
• б) A circle with center O and points A, B, C forming a triangle ABC. All vertices are on the circle, making it an inscribed triangle. A line segment from O to B is a radius. Angle ABC is an inscribed angle.
• в) Similar to figure б), with points A, B, C on the circle. Triangle ABC is inscribed. Angle ABC is an inscribed angle.
• г) A circle with center O and points A, B, C. A chord AC is drawn, and a line segment from O to A is a radius. Another line segment connects O to C, also a radius. Angle ABC appears to be an inscribed angle.
• д) A circle with center O and points A, B, C. A chord AC is drawn. A line segment connects O to A. A horizontal line passes through B and C. Angle ABC is formed by a chord AB and a chord BC.
• е) A circle with center O and points A, B, C. A line segment AB passes through the center O, making it a diameter. A point C is on the circle. Angle ABC appears to be inscribed, with one side being a diameter.
• ж) A circle with center O and points A, B, C. A line segment AB is a chord. A line segment BC is a chord. A horizontal line passes through C and appears tangent to the circle at C, or is a chord. Angle ABC is an inscribed angle.

The image also contains the heading "Рис. 267" (Figure 267) and "§3 Вписанная и описанная окружности" (Inscribed and circumscribed circles).