In the triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
let
![B'](/media/m/a/1/a/a1a88eb7f35fee4f41c66bfb0c902f51.png)
and
![C'](/media/m/0/0/1/001d1a1af4c90ceda662e79e88845742.png)
be the midpoints of the sides
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
and
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
respectively and
![H](/media/m/4/c/0/4c0872a89da410a25f00b86366efece7.png)
the foot of the altitude passing through the vertex
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
. Prove that the circumcircles of the triangles
![AB'C'](/media/m/3/8/4/384f3919c233da9fabbbe2aa90b0bd0d.png)
,
![BC'H](/media/m/e/6/e/e6e7876da13041dc70d4782cb844b632.png)
, and
![B'CH](/media/m/c/1/6/c166b779ee9d3f4835460b7a588d7163.png)
have a common point
![I](/media/m/3/8/6/38689d6affa9ba35368ca4d3d76ea147.png)
and that the line
![HI](/media/m/a/f/b/afbc6f3ee5408c1cdfb4168f4af5db10.png)
passes through the midpoint of the segment
%V0
In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$,$BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'.$