A quiet collection of problems, proofs, and diagrams devoted to the structure and beauty of elementary Euclidean geometry.https://elementarygeometry.org/enhttps://elementarygeometry.org/problems/crux-5021/https://elementarygeometry.org/problems/crux-5021/A direct-similarity generalization proved with Pappus's theorem and a radical axis, presented without reproducing the original problem statement.Mon, 10 Aug 2026 00:00:00 GMThttps://elementarygeometry.org/problems/crux-5031/https://elementarygeometry.org/problems/crux-5031/A concise incircle-inversion proof using contact-chord midpoints and a cyclic quadrilateral, presented without reproducing the original problem statement.Mon, 10 Aug 2026 00:00:00 GMThttps://elementarygeometry.org/problems/crux-5088/https://elementarygeometry.org/problems/crux-5088/A concise angle-chasing proof through tangency and cyclic quadrilaterals, presented without reproducing the original problem statement.Mon, 10 Aug 2026 00:00:00 GMThttps://elementarygeometry.org/problems/crux-5098/https://elementarygeometry.org/problems/crux-5098/A unit-circle complex proof that reduces two geometric conditions to the same algebraic equation, presented without reproducing the original problem statement.Mon, 10 Aug 2026 00:00:00 GMThttps://elementarygeometry.org/problems/crux-5109/https://elementarygeometry.org/problems/crux-5109/A synthetic fixed-point argument and a projective generalization, presented without reproducing the original problem statement.Mon, 10 Aug 2026 00:00:00 GMT