Proof:

Given: Triangle ABC with altitude AD bisecting BC

To prove: Triangle ABC is isosceles

Proof:

Step 1: Draw the figure

First, draw a triangle ABC with the given altitude AD bisecting the base BC.

Step 2: Let's assume the triangle is not isosceles

Assume that triangle ABC is not isosceles. This means that the lengths of the sides AB, BC, and CA are all different.

Step 3: Construct the perpendiculars

Construct perpendiculars from point B and C to the altitude AD, meeting at points E and F respectively.

Step 4: Length of AD

Since AD is an altitude, it is perpendicular to the base BC. Therefore, angle BAD and angle CAD are right angles.

Step 5: Length of BD and CD

Since AD bisects BC, BD and CD are equal in length.

Step 6: Length of BE and CF

Since BE and CF are perpendiculars to AD, they are equal in length.

Step 7: Triangle BED and CFD

Triangle BED and CFD are right-angled triangles. By the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Therefore, (BD)^2 + (BE)^2 = (CD)^2 + (CF)^2

But, BD = CD and BE = CF

So, (BD)^2 + (BE)^2 = (CD)^2 + (CF)^2 simplifies to (BD)^2 = (CD)^2

Step 8: Conclusion

From step 7, we can conclude that BD = CD.

Since BD = CD, triangle ABC is isosceles because two sides (BD and CD) are equal in length.

Therefore, our assumption in step 2 is false, and triangle ABC is indeed isosceles.

Hence, we have proved that if only altitude AD bisects BC, then triangle ABC is isosceles.