Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.
1. **Identify the given points and their properties:**
   - \( I \) is the incenter of the triangle \( \triangle ABC \).
   - \( M \) is the centroid of the triangle \( \triangle ABC \).
   - \( H \) is the orthocenter of the triangle \( \triangle ABC \).

2. **Use Euler's theorem:**
   Euler's theorem states that in any triangle, the orthocenter \( H \), the centroid \( M \), and the circumcenter \( O \) are collinear, and the distance between the centroid and the orthocenter is twice the distance between the centroid and the circumcenter:
   \[
   OH = 3OM
   \]
   Given \( H \) and \( M \), we can find \( O \) using the relation:
   \[
   O = M + \frac{1}{3}(H - M)
   \]

3. **Find the excenter \( I_A \):**
   The excenter \( I_A \) opposite to vertex \( A \) can be found using the properties of the triangle. Since \( I \) is the incenter, the excenter \( I_A \) is the intersection of the external angle bisectors of \( \angle B \) and \( \angle C \).

4. **Use the harmonic division property:**
   Since \( BI \) is the bisector of \( \angle HBO \), the points \( O, H, I, I_A \) form a harmonic division:
   \[
   (OH, II_A) = -1
   \]
   This property helps in determining the relative positions of these points.

5. **Find the midpoint \( T \) of \( II_A \):**
   Let \( T \) be the midpoint of \( II_A \). The coordinates of \( T \) can be found as:
   \[
   T = \frac{I + I_A}{2}
   \]

6. **Determine the points \( B \) and \( C \):**
   Points \( B \) and \( C \) are the intersections of:
   - The circle with center \( O \) and radius \( OT \).
   - The circle with center \( T \) and radius \( IT \).

7. **Find the point \( A \):**
   Point \( A \) is the intersection of the line \( OH \) and the circle with center \( O \) and radius \( OT \).

By following these steps, we can restore the isosceles triangle \( \triangle ABC \).

The final answer is the restored isosceles triangle \( \boxed{ \triangle ABC } \).