The smaller the legs of an isosceles triangle with a constant angle between legs the smaller the base

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Let ABC be an isosceles triangle where the legs are AB and AC. How can you prove (rigorously) that the smaller AB and AC get the smaller BC gets? (In other words, BC is minimal when AB and AC are minimal)

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Isosceles triangles with the same angle between the legs are similar because all the angles are the same. Similarity shows the sides are in proportion, so the smaller the legs, the smaller the base.