Let

be an acute-angled triangle with

. Let

be the orthocenter of triangle

, and let

be the midpoint of the side

. Let

be a point on the side

and

a point on the side

such that

and the points

,

,

are on the same line. Prove that the line

is perpendicular to the common chord of the circumscribed circles of triangle

and triangle

.
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Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.