In the provided image, there are several geometry problems related to triangles. Please analyze each problem and provide a structured solution or the requested information for each. Start with problem 1, then proceed to problem 2, and so on. For each problem, identify the type of triangle, any given measurements, and what needs to be found. If a solution is calculable, show the steps. If it's a statement or definition, present it clearly. The table title mentions 'Area of a triangle' and 'O is the center of the circle'.

Problem 1:

The image displays a triangle with angles labeled. One angle is 135 degrees. There are lengths indicated as 3 and 8. Point D is on side AC, and angle BDC is a right angle. It appears to be asking for the area of triangle ABC, but the exact question is unclear from the cropped image.

Problem 2:

This problem shows a triangle ABC with sides labeled 10, 10, and 8. It appears to be an isosceles triangle. The question is likely to find the area or other properties of this triangle.

Problem 3:

This is a triangle with sides labeled 9, 8, and C. Angle A is 30 degrees, and there is a circle inscribed within the triangle, with its center labeled O and radius 3. This is likely related to finding the area of the triangle using properties of incircles.

Problem 4:

This problem shows a triangle with sides labeled 13, 12, and B. Point C is a right angle. This is a right-angled triangle, likely asking for the length of side B or the area.

Problem 5:

This is a triangle with sides labeled 14, 13, and B. There is an inscribed circle with center O and radius 2. This problem might involve finding the area or other properties using the inradius formula.

Problem 6:

This problem depicts a triangle with sides labeled 9x, 8x, and E. There is an inscribed circle with center O and radius 24. The segment from vertex B to the point of tangency on side AB is labeled 9x, and the segment from vertex A to the point of tangency on side AC is labeled 8x. This problem is likely about finding the value of x or the area.

Problem 7:

This shows a triangle inscribed in a circle. Sides are labeled 5, 6, and B. Point O is the center of the circle. This is likely related to finding properties of a triangle inscribed in a circle, possibly the circumradius or area.

Problem 8:

This is a triangle with sides labeled 12, M, and B. Point K is on side AB, and there is an inscribed circle with center O. This problem is likely related to finding lengths of segments or the area using properties of tangents from a vertex to an incircle.

Problem 9:

This is a triangle with sides labeled 15, D, and C. There is an inscribed circle with center O and radius 2. This problem seems similar to problem 5, involving an incircle and side lengths.

Problem 10:

This problem shows a triangle ABC. Side AC is divided into segments AD = 15 and DC = 20. Angle C is bisected by a line segment from C to a point D on AB. This is likely a problem involving the angle bisector theorem or properties of angle bisectors.

Problem 11:

This problem depicts a triangle with sides labeled 12, E, and B. Point D is on side AC. There is an inscribed circle with center O. This appears similar to other incircle problems.

Problem 12:

This problem states "Дано: \(\angle\) A = \(\angle\) C". This implies that the triangle is isosceles with sides opposite to these angles being equal (AB = BC). The problem is likely to find other properties or the area of this isosceles triangle.

Problem 13:

This problem states "Дано: P = 84". This means the perimeter of the triangle is 84. This information would be used in conjunction with other given side lengths or properties to solve for unknown values.

General Observation:

The table is titled "Table 9.9. Area of a triangle". The general instruction is "Find the area of triangle ABC". Many problems involve incircles, suggesting formulas related to area and inradius (Area = rs, where r is the inradius and s is the semi-perimeter) will be useful. Some problems involve right-angled triangles, where the Pythagorean theorem and standard area formula (1/2 * base * height) are applicable. Isosceles triangles and angle bisectors are also present.