Let ABCDE be a convex pentagon and extend the sides to form a pentagram A_{1}B_{1}C_{1}D_{1}E_{1}. Construct the green circumcircles of the five yellow triangles. Then the five new points, A_{2}, B_{2}, C_{2}, D_{2}, E_{2} (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.

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.

GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus application, intended for teachers and students. Many parts of GeoGebra have been ported to HTML5.

Geometry Problems

Open Problems

Visual Index

Ten problems: 1411-1420

All Problems

Pentagon

Triangle

Circle

Triangle Centers

Circumcircle

Intersecting Circles

Concyclic
Points

Classical Theorems

GeoGebra

HTML5 and Dynamic Geometry

iPad Apps

View or Post a solution