Using only a compass and a ruler, reconstruct triangle $ABC$ given the following three points: point $M$ the intersection of its medians, point $I$ is the center of its inscribed circle and the point $Q_a$ is touch point of the inscribed circle to side $BC$.
1. **Identify the Nagel Point \( N \)**:
   - Given points \( I \) (incenter), \( M \) (centroid), and \( Q_a \) (touch point of the incircle with side \( BC \)).
   - The centroid \( M \) divides the segment \( NI \) in the ratio \( 2:1 \) internally. This means \( N \) is the Nagel point of \( \triangle ABC \).
   - To find \( N \), extend the line \( IM \) such that \( M \) divides \( NI \) in the ratio \( 2:1 \).

2. **Construct the Incircle \( \omega \)**:
   - Use the compass to draw a circle centered at \( I \) with radius \( IQ_a \). This circle is the incircle of \( \triangle ABC \).

3. **Reflect \( Q_a \) across \( I \)**:
   - Reflect \( Q_a \) across \( I \) to get \( Q'_a \). This can be done by constructing the perpendicular bisector of \( IQ_a \) and finding the point on the other side of \( I \) at the same distance as \( Q_a \).

4. **Draw Line \( \ell \) (Side \( BC \))**:
   - Draw the line perpendicular to \( IQ_a \) at \( Q_a \). This line is the side \( BC \) of \( \triangle ABC \). Call this line \( \ell \).

5. **Find Intersection Point \( J \)**:
   - Draw the line \( NQ'_a \) and let it intersect \( \ell \) at point \( J \).

6. **Determine Midpoint \( D \) of \( JQ_a \)**:
   - Find the midpoint \( D \) of segment \( JQ_a \). Since \( D \) is the midpoint of \( JQ_a \), it is also the midpoint of side \( BC \).

7. **Locate Vertex \( A \)**:
   - Extend the line \( DM \) and the line \( JN \). The intersection of these two lines is point \( A \).

8. **Construct Tangents from \( A \) to \( \omega \)**:
   - Draw the two tangents from \( A \) to the incircle \( \omega \). These tangents will touch the circle at points \( B \) and \( C \), which lie on line \( \ell \).

The final answer is the construction of \( \boxed{ \triangle ABC } \) using the given points \( M \), \( I \), and \( Q_a \).