Dynamic Geometry 1477: Miquel's Pentagram Theorem, Pentagon, Triangle, Circumcircles, Concyclic Points, Step-by-step Illustration

Let ABCDE be a convex pentagon and extend the sides to form a pentagram A1B1C1D1E1. Construct the green circumcircles of the five yellow triangles. Then the five new points, A2, B2, C2, D2, E2 (blue points) resulting from the intersection of each pair of adjacent circles are concyclic (lie on the same circle). See dynamic diagram.

Auguste Miquel (France, Nantua, College des Castres.) published this beautiful theorem in Journal de Mathematiques Pures et Appliquees (Liouville s Journal) Tome Troisieme, Paris 1838.


Static Diagram of Miquel's Pentagram Theorem

Problem 1477 Miquel's Pentagram Theorem, Pentagon, Triangle, Circumcircles, Concyclic Points, Step-by-step Illustration, iPad Apps


Poster of the Miquel's Pentagram Theorem using iPad Apps

Dynamic Geometry 1477: Miquel's Pentagram Theorem, Pentagon, Triangle, Circumcircles, Concyclic Points, Step-by-step Illustration Using GeoGebra, iPad Apps

Classroom Resource:
Interactive step-by-step animation using GeoGebra

This step-by-step interactive illustration was created with GeoGebra.

  • To explore (show / hide): click/tap a check box.
  • To stop/play the animation: click/tap the icon in the lower left corner.
  • To go to first step: click/tap the "Go to step 1" button.
  • To manipulate the interactive figure: click/tap and drag the blue points or figures.

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