Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an acute-angled triangle such that
![\angle ABC<\angle ACB](/media/m/2/1/c/21c59944e1abb5ef6fb6fc735bb164b4.png)
, let
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
be the circumcenter of triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
, and let
![D=AO\cap BC](/media/m/b/4/4/b4416ca7f1b63a98be3fd9d7166728ca.png)
. Denote by
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
the circumcenters of triangles
![ABD](/media/m/a/5/4/a548bc577543629d304ecba1a042f910.png)
and
![ACD](/media/m/0/b/1/0b171034d79122bd02f64bc8f6ae94dd.png)
, respectively. Let
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
be a point on the extension of the segment
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
beyound
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
such that
![AG=AC](/media/m/f/e/6/fe6f81cbf3d204ae31357a8f8d2a2b56.png)
, and let
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
be a point on the extension of the segment
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
beyound
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
such that
![AH=AB](/media/m/4/7/3/473a78bd4efadacf9770e98f0fff67c3.png)
. Prove that the quadrilateral
![EFGH](/media/m/5/4/6/546f6a8c4c38499f3e56b70541e9470d.png)
is a rectangle if and only if
![\angle ACB-\angle ABC=60^\circ](/media/m/0/6/b/06b4dd4ff630f0567f3d661c090d6950.png)
.
comment
Official version Let
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
be the circumcenter of an acute-angled triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
with
![{\angle B<\angle C}](/media/m/2/7/1/271760487a7fd7fc0964a6c361c53ace.png)
. The line
![AO](/media/m/d/9/3/d93e4e1fde6437bd5210d0a50abb3ca8.png)
meets the side
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
at
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
. The circumcenters of the triangles
![ABD](/media/m/a/5/4/a548bc577543629d304ecba1a042f910.png)
and
![ACD](/media/m/0/b/1/0b171034d79122bd02f64bc8f6ae94dd.png)
are
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
and
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
, respectively. Extend the sides
![BA](/media/m/f/3/e/f3ee5efe9b25bd27cd3ada1235d36017.png)
and
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
beyond
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, and choose on the respective extensions points
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
and
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
such that
![{AG=AC}](/media/m/a/9/2/a92199f3547e682d14b0419bb905b3f7.png)
and
![{AH=AB}](/media/m/c/3/a/c3a3fe4a582a0093ead69eda16c8dad1.png)
. Prove that the quadrilateral
![EFGH](/media/m/5/4/6/546f6a8c4c38499f3e56b70541e9470d.png)
is a rectangle if and only if
![{\angle ACB-\angle ABC=60^{\circ }}](/media/m/7/a/1/7a1584acc45d5d9888e0d8b2db1f7e32.png)
.
Edited by orl.
%V0
Let $ABC$ be an acute-angled triangle such that $\angle ABC<\angle ACB$, let $O$ be the circumcenter of triangle $ABC$, and let $D=AO\cap BC$. Denote by $E$ and $F$ the circumcenters of triangles $ABD$ and $ACD$, respectively. Let $G$ be a point on the extension of the segment $AB$ beyound $A$ such that $AG=AC$, and let $H$ be a point on the extension of the segment $AC$ beyound $A$ such that $AH=AB$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if $\angle ACB-\angle ABC=60^\circ$.
comment
Official version Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.
Edited by orl.