Cho lăng trụ tam giác ABC.A'B'C' có đáy ABC là tam giác đều cạnh bằng a, hình chiếu của A' lên mặt phẳng (ABC) trùng với trọng tâm G của tam giác ABC

To tackle the problem, we'll break it down step by step as per the given tasks.

### a) For the tetrahedron \( ABC.A'B'C' \):
1. **Given**:
- Triangle \( ABC \) is an equilateral triangle with side length \( a \).
- The projection of \( A' \) onto the plane \( (ABC) \) coincides with the centroid \( G \) of triangle \( ABC \).

2. **Find the area of the triangular base \( ABC \)**:
- The area \( S_{ABC} \) of an equilateral triangle can be calculated using the formula:
\[
S = \frac{\sqrt{3}}{4} a^2
\]

3. **Determine the height from the centroid \( G \) to the plane**:
- There’s no additional information yet on the height \( h \) from \( G \) to \( A' \) and it’s required to find the full volume of the tetrahedron.

4. **Calculate the volume of the tetrahedron using the height h**:
- The volume \( V \) of the tetrahedron can be calculated using the formula:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times S_{ABC} \times h
\]

5. **Final Answer Part a**:
- You would plug in \( S_{ABC} \) and \( h \) afterward to get numerical values.

### b) For the triangular pyramid \( S.ABC \):
1. **Given**:
- Height \( SA \) is perpendicular to \( BC \).
- \( K \) is the midpoint of \( BC \).
- \( KB = KC = a; \angle KBC = 60^\circ \).
- \( A \) is above the plane \( (ABC) \) at an angle of \( 45^\circ \) to plane \( (SKC) \).

2. **Find the distance from \( A \) to the plane \( (SKC) \)**:
- You need to calculate the height from \( A \) above \( G \) and find the projection.

3. **Geometry Knowledge**:
- Using the angle \( 45^\circ \) and triangle properties, use trigonometric ratios to find the height or distance you're seeking.

### c) For the tetrahedron \( ABCD \) with the angles formed:
1. **Given**:
- \( D' \) is the intersection of the plane \( \alpha \) with segments \( D'A, D'B, D'C \).
- Calculate the distances \( D'M, D'N, D'P \) which are the distances from point \( O \) to respective vertices.

2. **Evaluate \( T \)**:
- Once you find those distances, proceed to calculate \( T \) using the harmonic mean formula given:
\[
T = \frac{1}{D'M \cdot D'P} + \frac{1}{D'P \cdot D'N} + \frac{1}{D'N \cdot D'M}
\]

3. **Final Computation**:
- Plug in all found distances to yield the final numerical result.

### Final Notes:
Remember to clearly outline each step of your calculations and verify any intermediate results to avoid potential errors!