Section Circumcenter2 .
Context `{TE:Tarski_euclidean}.
(** # <script type="text/javascript" language="javascript" src=" http://www.geogebra.org/web/4.2/web/web.nocache.js"></script><article class="geogebraweb" data-param-width="800" data-param-height="600" data-param-showResetIcon="false" data-param-enableLabelDrags="false" data-param-showMenuBar="false" data-param-showToolBar="false" data-param-showAlgebraInput="false" enableLabelDrags="true" data-param-ggbbase64="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"></article> # **)

Lemma circumcenter_perp_all : forall A B C A' B' C' G,
A<>B -> B<>C -> A<>C -> G <> A' -> G <> B' -> G <> C' ->
is_circumcenter G A B C ->
Midpoint A' B C ->
Midpoint B' A C ->
Midpoint C' A B ->
Perp_bisect G A' B C /\ Perp_bisect G B' A C /\ Perp_bisect G C' A B.