Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point. 

[i](The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)[/i]
1. **Given Data and Initial Construction:**
   - Let the given vertex be \( A \).
   - We are also given the circumcenter \( O \) and the Lemoine point \( L_e \).
   - Construct the circumcircle \(\Omega\) of \(\triangle ABC\) with center \( O \) and radius \( OA \).

2. **Intersection of Line \( AL_e \) with Circumcircle:**
   - Draw the line \( AL_e \) and let it intersect the circumcircle \(\Omega\) at points \( A \) and \( P \).

3. **Constructing Tangents and Apollonius Circle:**
   - Construct the tangents from points \( A \) and \( P \) to the circumcircle \(\Omega\). Let these tangents intersect at point \( Q \).
   - Point \( Q \) is the center of the \( A \)-Apollonius circle.

4. **Isodynamic Points and Pedal Triangles:**
   - It is known that the isodynamic points lie on the line \( OL_e \) and their pedal triangles are equilateral.
   - Construct the circle with center \( Q \) and radius \( QA \). Let this circle intersect the line \( OL_e \) at points \( A_p \) and \( A_p' \).

5. **Inversion and Equilateral Triangle Construction:**
   - Choose one of the points \( A_p \) or \( A_p' \), say \( A_p \).
   - Invert through \( A_p \) with center \( A_p \) and radius \( AA_p \). Since the pedal triangle of the isodynamic point is equilateral, the image of \(\triangle ABC\) under this inversion will also be equilateral.
   - Construct the equilateral triangle given one of its vertices \( A \) and its circumcircle \(\Omega_1\).

6. **Trisecting the Circumcircle:**
   - Trisect the circumcircle \(\Omega_1\) into three equal arcs, two of which have \( A \) as an endpoint.
   - This gives us the vertices \( B_1 \) and \( C_1 \) of the equilateral triangle \(\triangle AB_1C_1\).

7. **Finding the Original Vertices:**
   - The original vertices \( B \) and \( C \) can be found by intersecting the lines \( A_pB_1 \) and \( A_pC_1 \) with the circumcircle \(\Omega\).
   - Ensure the order is \( A_p - B_1 - B \) and \( A_p - C_1 - C \).

8. **Conclusion:**
   - We have successfully restored the triangle \(\triangle ABC\) using the given vertex \( A \), the circumcenter \( O \), and the Lemoine point \( L_e \).

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