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In a triangle $\Delta ABC$,suppose E is the midpoint of AC,F is the midpoint of AB,$BE\bot CF$,point O,I,H are the circumcenter, incenter, and orthocenter of this triangle, respectively.

"circle I" denotes the inscribed circle of the triangle ABC.

Prove: If point H is on the "circle I",then point O is on the "circle I" too.

I tried to use the knowledge of Euler's circle to solve this question，and I connect line segment OE,and I suppose $BE \bigcap CF=G$,where G is the center of gravity of $\Delta ABC$,but I don't know what to do next.

In a triangle $\Delta ABC$,suppose E is the midpoint of AC,F is the midpoint of AB,$BE\bot CF$,point O,I,H are the circumcenter, incenter, and orthocenter of this triangle, respectively.

"circle I" denotes the inscribed circle of the triangle ABC.

Prove: If point H is on the "circle I",then point O is on the "circle I" too.

I tried to use the knowledge of Euler's circle to solve this question，and I connect line segment OE,and I suppose $BE \bigcap CF=G$,where G is the center of gravity of $\Delta ABC$,but I don't know what to do next. Can anyone provide an answer?

In a triangle $\Delta ABC$,suppose E is the midpoint of AC,F is the midpoint of AB,$BE\bot CF$,point O,I,H are the circumcenter, incenter, and orthocenter of this triangle, respectively.

circle I means the inscribed circle of the triangle ABC. Prove: If point H is on circle I,then point O is on circle I too.

I tried to use the knowledge of Euler's circle to solve this question，and I connect line segment OE,and I suppose $BE \bigcap CF=G$,where G is the center of gravity of $\Delta ABC$,but I don't know what to do next.

In a triangle $\Delta ABC$,suppose E is the midpoint of AC,F is the midpoint of AB,$BE\bot CF$,point O,I,H are the circumcenter, incenter, and orthocenter of this triangle, respectively.

"circle I" denotes the inscribed circle of the triangle ABC.

Prove: If point H is on the "circle I",then point O is on the "circle I" too.

I tried to use the knowledge of Euler's circle to solve this question，and I connect line segment OE,and I suppose $BE \bigcap CF=G$,where G is the center of gravity of $\Delta ABC$,but I don't know what to do next.

In a triangle $\Delta ABC$,suppose E is the midpoint of AC,F is the midpoint of AB,$BE\bot CF$,O,I,H are the circumcenter, incenter, and orthocenter of this triangle, respectively.

Prove: If H is on circle I,then O is on circle I too.

I tried to use the knowledge of Euler's circle to solve this question，and I connect line segment OE,and I suppose $BE \bigcap CF=G$,where G is the center of gravity of $\Delta ABC$,but I don't know what to do next.

In a triangle $\Delta ABC$,suppose E is the midpoint of AC,F is the midpoint of AB,$BE\bot CF$,point O,I,H are the circumcenter, incenter, and orthocenter of this triangle, respectively.

circle I means the inscribed circle of the triangle ABC. Prove: If point H is on circle I,then point O is on circle I too.

I tried to use the knowledge of Euler's circle to solve this question，and I connect line segment OE,and I suppose $BE \bigcap CF=G$,where G is the center of gravity of $\Delta ABC$,but I don't know what to do next.