1. **Problem Statement:** Show that if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle. 2. **Recall the properties:** - A parallelogram has opposite sides equal and opposite angles equal. - The diagonals of a parallelogram bisect each other. - A rectangle is a parallelogram with all angles equal to 90 degrees. 3. **Let the parallelogram be ABCD with diagonals AC and BD.** 4. **Use the midpoint formula:** Since diagonals bisect each other, let O be the midpoint of both AC and BD. 5. **Express the diagonals as vectors:** Let $ \vec{AB} = \vec{u} $ and $ \vec{AD} = \vec{v} $. Then, $ \vec{AC} = \vec{u} + \vec{v} $ and $ \vec{BD} = \vec{v} - \vec{u} $. 6. **Given diagonals are equal:** $$|\vec{AC}| = |\vec{BD}|$$ $$\Rightarrow |\vec{u} + \vec{v}| = |\vec{v} - \vec{u}|$$ 7. **Square both sides:** $$|\vec{u} + \vec{v}|^2 = |\vec{v} - \vec{u}|^2$$ $$ (\vec{u} + \vec{v}) \cdot (\vec{u} + \vec{v}) = (\vec{v} - \vec{u}) \cdot (\vec{v} - \vec{u}) $$ 8. **Expand dot products:** $$ \vec{u} \cdot \vec{u} + 2 \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{v} = \vec{v} \cdot \vec{v} - 2 \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{u} $$ 9. **Simplify:** $$ 2 \vec{u} \cdot \vec{v} = -2 \vec{u} \cdot \vec{v} $$ $$ 4 \vec{u} \cdot \vec{v} = 0 $$ $$ \Rightarrow \vec{u} \cdot \vec{v} = 0 $$ 10. **Interpretation:** The dot product of $ \vec{u} $ and $ \vec{v} $ is zero, meaning $ \vec{u} $ is perpendicular to $ \vec{v} $. 11. **Conclusion:** Since adjacent sides are perpendicular, the parallelogram has right angles and is therefore a rectangle. **Final answer:** If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.