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 enter image description here

  • 3
    $\begingroup$ 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? $\endgroup$ Dec 18, 2022 at 15:56
  • $\begingroup$ It seems that $D,E$ are wrongly labeled in the picture. $\endgroup$
    – user376343
    Dec 18, 2022 at 19:58
  • $\begingroup$ 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? $\endgroup$
    – nickgard
    Dec 19, 2022 at 19:24
  • $\begingroup$ Your drawing is misleading to think that the intersection is H. What is G? where are the other points? $\endgroup$
    – Moti
    Dec 21, 2022 at 6:08


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