
elegant
Describes a person, clothing, or movement that is tasteful, graceful, and stylish.
Sie trug ein sehr elegantes Abendkleid.
She wore a very elegant evening gown.
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Describes a person, clothing, or movement that is tasteful, graceful, and stylish.
Sie trug ein sehr elegantes Abendkleid.
She wore a very elegant evening gown.

Describes a solution, idea, or proof that works in a clever, simple, and very effective way.
Der Programmierer fand eine elegante Lösung für das komplexe Problem.
The programmer found an elegant solution to the complex problem.
The meaning of 'elegant' depends strongly on the noun. If it refers to people, clothing, or movements, it means 'stilvoll' (stylish). If it refers to ideas, solutions, or methods, it means 'klug und einfach' (clever and simple).
Die Tänzerin war für ihre eleganten Bewegungen bekannt.
The dancer was known for her elegant movements.
In connection with appearance and style, 'elegant' is often combined with clothing, manner, or movements. Typical combinations are 'elegant gekleidet sein' (to be elegantly dressed), 'ein elegantes Auto' (an elegant car), or 'elegant aussehen' (to look elegant).
Er kam in einem eleganten Anzug zur Preisverleihung.
He came to the awards ceremony in an elegant suit.
In the abstract meaning, 'elegant' is often used in academic or technical contexts. Common combinations are 'eine elegante Lösung' (an elegant solution), 'eine elegante Methode' (an elegant method), or 'eleganter Code' (elegant code).
Die Ingenieurin entwickelte eine elegante Methode, um Energie zu sparen.
The engineer developed an elegant method for saving energy.
brilliant / awesome
Genial and elegant can both describe a clever, successful solution. Genial mainly emphasizes exceptional creativity or inventiveness. Elegant is appropriate when the solution is also simple, clear, and effective.

Die Lösung ist elegant, weil sie mit drei einfachen Schritten zum Ziel führt.
The solution is elegant because it reaches the goal in three simple steps.
Die Lösung ist genial, weil sie auf einer völlig neuen Idee beruht.
The solution is brilliant because it is based on a completely new idea.