# prove that $BC$ is perpendicular to $MN$

Given is an acute triangle $$ABC$$ where $$H$$ is the intersection of the three altitudes of the triangle. Points $$B_1$$ and $$C_1$$ respectively lie on the outer circle of triangle ABC such that $$BB_1$$ and $$CC_1$$ are the diameters of the outer circle of triangle $$ABC$$. Points $$D$$ and $$E$$ are the midpoints of sides $$AB$$ and sides $$AC$$, respectively. If $$DC$$ and $$BE$$ intersect at point $$M$$, while $$B_1D$$ and $$C_1E$$ intersect at point $$N$$, prove that $$BC$$ is perpendicular to $$MN$$.

Is my geometry drawing correct? anyone can help me for this problem solve? Thank you

• It is difficult to see which points you have labelled as $B_{1}, C_{1}$ etc. Have you tried to label all points mentioned in the question? Dec 18, 2022 at 15:56
• It seems that $D,E$ are wrongly labeled in the picture. Dec 18, 2022 at 19:58
• Point $H$ is defined in the first sentence. I could comment that point $H$ in the diagram should not be on the circumcircle but inside the triangle at the intersection of the three altitudes; however, this is irrelevant because point $H$ and the three altitudes of $ABC$ are not used at all in the rest of the problem. Is there a problem with the transcription of this problem? Dec 19, 2022 at 19:24
• Your drawing is misleading to think that the intersection is H. What is G? where are the other points?
– Moti
Dec 21, 2022 at 6:08