Proof:

Given, two equiangular triangles ABC and DEF.

Let the corresponding sides of the triangles be AB, BC, and AC for triangle ABC and DE, EF, and DF for triangle DEF.

Let the medians of the triangles be AX, BY, and CZ for triangle ABC and DG, EH, and FI for triangle DEF.

First Step:

We need to prove that the two triangles are similar to each other.

Since the two triangles are equiangular, the corresponding angles in both triangles are equal.

Therefore, ∠A = ∠D, ∠B = ∠E, and ∠C = ∠F.

Hence, the two triangles ABC and DEF are similar to each other by the Angle-Angle (AA) similarity criterion.

Second Step:

Now, we need to prove that the ratio of the corresponding sides is the same as the ratio of the corresponding medians.

Let us consider the ratio of the corresponding sides:

AB/DE = BC/EF = AC/DF

Since the triangles are similar, we can write:

AB/DE = BC/EF = AC/DF = k (say)

Now, let us consider the medians of the triangles.

The medians of the triangle ABC are AX, BY, and CZ.

The medians of the triangle DEF are DG, EH, and FI.

Let us consider the ratio of the corresponding medians:

AX/DG = BY/EH = CZ/FI

We know that the medians of the triangle divide each side into two equal parts.

Therefore, we can write:

AX = ½ AB, BY = ½ BC, CZ = ½ AC

DG = ½ DE, EH = ½ EF, FI = ½ DF

Substituting these values, we get:

AX/DG = (½ AB)/(½ DE) = AB/DE

BY/EH = (½ BC)/(½ EF) = BC/EF

CZ/FI = (½ AC)/(½ DF) = AC/DF

Since AB/DE = BC/EF = AC/DF = k, we can write:

AX/DG = BY/EH = CZ/FI = k

Therefore, the ratio of the corresponding sides is the same as the ratio of the corresponding medians.

Hence, Proved.