Theorems of the 20th Century

Are all the beautiful facts of geometry were found in Ancient Greece, and we just have to study their evidence? Not at all! Let's remember the theorems of Morley, Thébault, Pompeiu as well as the circles of Conway and Van Lamoen 0:00 — Intro and nine-point circle 0:32 — Morley's trisector theorem 1:05 — Conway circle theorem" 1:25 — Pompeiu's theorem 2:40 — Napoleon's theorem 3:00 — Thébault's theorem 4:20 — Van Aubel's theorem 4:40 — Van Lamoen circle Hint to the Pompeu's theorem. Note that ∠APB = ∠ACB = 60° as inscribed Hint to the Thébault's theorem. The angle between a side of a square and its diagonal is 45°. Let ∠BCD= γ, then ∠MCN=90°+ γ. ∠ABC=180°–∠BCD=180°–γ due to the property of one-sided angles. On the other hand, ∠ABC=360°–(∠CBM+∠MBL+∠LBA)=270°–∠MBL. Hence ∠MBL=270°–∠ABC=270°–(180°– γ )=90°+ γ=∠MCN, Q.E.D. MORE BEAUTIFUL GEOMETRY 1) Geometric Interpretations:    • The Sum of the First Cubes, Squares and More   2) Fagnano's Problem:    • Geometry in Animations. Fagnano's Problem   3) Fractal Geometry:    • The Beauty of Fractal Geometry (#SoME2)