Mathematical equations and geometric figures on a chalkboard, illustrating concepts in trigonometry and triangle properties.

- The image displays a collection of geometric formulas related to triangles, primarily focusing on right triangles and general triangles.
- Key formulas include the Pythagorean theorem (a² + b² = c²), area calculations (S = ab/2 for right triangles, S = 1/2 * base * height, S = abc/(4R), S = pr), and relationships involving medians (m_b = 1/2 * √(2a² + 2c² - b²)), altitudes (h_b = √(a² - b²/4)), and trigonometric functions (a = c sinα, a = c cosβ).
- It also shows formulas for the circumradius (R = c/2 for right triangles) and inradius (r = (a + b + c)/2 for right triangles, though this is incorrect; it should be r = (a + b - c)/2 for right triangles or r = S/p for any triangle where p is the semi-perimeter).
- The angle relationship α = π/2 - β indicates complementary angles in a right triangle.
- The formula AD/DC = AB/BC suggests the Angle Bisector Theorem.
- The notation S = 1/2 ab sinγ applies to any triangle with sides a and b and included angle γ.
- The formula m_c = c/2 implies that the median to the hypotenuse in a right triangle is half the hypotenuse.
- Some formulas are incorrectly written or lack context (e.g., S = b h_b / 2 - a² sinβ / 2 is not a standard area formula; the correct form is S = 1/2 * b * h_b or S = 1/2 * a * b * sinγ).