Explanation:
The altitude to the base of an isosceles triangle meets the base at right angles, dividing the triangle into two right triangles. The hypotenuse of each of these triangles is one of the congruent sides of the isosceles triangle, so the hypotenuses are congruent. The altitude is congruent to itself, so comprises congruent legs of the right triangles.
The right triangles are congruent to each other by the HL congruence theorem. The remaining side of each of the right triangles is part of the base of the original isosceles triangle. Those sides are congruent by CPCTC, so the altitude meets the base at its midpoint. Hence the altitude to the base is a median of the isosceles triangle.
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Here's the same argument with vertices defined.
Define isosceles triangle ABC such that AB≅AC. Define altitude AD⊥BC. Then ΔADB and ΔADC are right triangles with AB≅AC and AD≅AD. Then ΔADB≅ΔADC by the HL theorem, and DB≅DC by CPCTC. Hence D is the midpoint of BC, and AD is a median.
