To solve this geometry problem, we need to follow the steps outlined in the question regarding triangle ABC and its properties. Here’s a layout of how to approach this:

1. **Given Information:**
- Triangle \( ABC \) is a right triangle at \( B \) with \( \angle C = 60^\circ \).
- The medians and points \( E, M, N, K \) are defined based on the triangle's sides.

2. **a) Proving \( \triangle ANM = \triangle BNAM \):**
- Use properties of triangles and congruence criteria (e.g., SAS or SSS) to show that the two triangles are congruent.

3. **b) Proving point \( E \) is equidistant from \( MB \) and \( BC \):**
- Apply the distance formula or properties of medians to establish the distances.

4. **c) Constructing point \( D \) on segment \( BE \) such that \( ED = \frac{1}{3} EB \):**
- Calculate the location of point \( D \) based on the given ratio and ensure that \( I \) is the midpoint of \( ME \).
- Verify the collinearity of points \( C, D, I \) using vector or coordinate geometry.

### Sketch:
1. **Begin with the Triangle:**
- Draw triangle \( ABC \) ensuring \( \angle ABC = 90^\circ \) and \( \angle C = 60^\circ \).

2. **Indicate Points and Lines:**
- Mark points \( M, N, E, K \) as specified.
- Construct the heights and medians as described.

3. **Label Distances Clearly:**
- Ensure distances corresponding to medians and points are labeled on your sketch.

By carefully executing these steps, you should be able to prove the required properties of the triangle and the related points. If you have specific measurements or values, incorporate them into your calculations for accurate results.