The problem requires proving that a triangle with two congruent altitudes is an isosceles triangle. Here's the solution and reasoning for the proof:

Proof

1. Given: A triangle $\triangle ABC$ has two congruent altitudes, say $h_1$ and $h_2$, dropped from vertices $A$ and $B$ to opposite sides $BC$ and $AC$, respectively.

2. Objective: Prove that $\triangle ABC$ is isosceles.

3. Strategy:

   • Use the overlapping triangles created by the altitudes $h_1$ and $h_2$.
   • Demonstrate that these triangles are congruent, which implies equality of certain sides or angles.

4. Analysis: Consider the overlapping right triangles $\triangle AHD$ and $\triangle BHE$, where:

   • $H$ is the foot of the altitude from $A$, so $AH \perp BC$.
   • $D$ is the foot of the altitude from $B$, so $BH \perp AC$.

5. Congruence by Rule:

   • In $\triangle AHD$ and $\triangle BHE$:

     • $AH = BH$ (Given: the two altitudes are congruent).
     • $\angle AHD = \angle BHE = 90^\circ$ (Definition of altitude).
     • $HD = HE$ (Shared segment from the orthocenter to the perpendicular feet).

     By the RHS (Right Angle-Hypotenuse-Side) congruence rule, $\triangle AHD \cong \triangle BHE$.

6. Implications:

   • From the congruence, $\angle A = \angle B$, proving $AB$ is equal to $AC$ by the property of isosceles triangles.

7. Conclusion: Since two angles (or corresponding sides) are equal, $\triangle ABC$ is isosceles.

Completing the Answer:

The congruence rule to fill in the blank in the problem is RHS (Right Angle-Hypotenuse-Side).

Would you like me to clarify any steps or provide further details on this proof?

Related Questions:

1. What does the RHS congruence rule state, and why is it applicable here?
2. Why are the overlapping triangles critical in proving the isosceles nature of the triangle?
3. How does the definition of altitude ensure a right angle in the triangles?
4. Can this proof be adapted for obtuse or right triangles?
5. What are other properties of isosceles triangles that could validate this conclusion?

Tip:

Always check which segments or angles are congruent when using overlapping triangles in proofs—they often simplify complex geometric relationships.