Problem: Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocenter $H$. Let $K$ be a point of $\Gamma$ on the other side of $\ol{BC}$ from $A$. Let $L$ be the reflection of $K$ across $\ol{AB}$, and let $M$ be the reflection of $K$ across $\ol{BC}$. Let $E$ be the second point of intersection of $\Gamma$ with the circumcircle of triangle $BLM$.  Show that the lines $KH$, $EM$ and $BC$ are concurrent.

Solution: \begin{center}
  \begin{asy}
    size(9cm);
    pair A = dir(110);
    pair B = dir(210);
    pair C = dir(330);
    draw(A--B--C--cycle);

    pair Ap = -B*C/A;
    pair Cp = -A*B/C;
    draw(A--Ap);
    draw(C--Cp);
    draw(unitcircle);

    pair K = dir(230);
    pair M = reflect(B,C)*K;
    pair blah = reflect(A,B)*K;
    pair L = blah;
    draw(M--K--L, dotted);

    pair E = IP(unitcircle, circumcircle(B, L, M));
    draw(circumcircle(B, L, M));

    pair H = A+B+C;
    draw(H--K, red);
    draw(E--Ap, red);
    draw(E--L, dotted);

    dot("$A$", A, dir(A));
    dot("$B$", B, dir(B));
    dot("$C$", C, dir(C));
    dot("$A'$", Ap, dir(Ap));
    dot("$C'$", Cp, dir(Cp));
    dot("$K$", K, dir(K));
    dot("$M$", M, dir(M));
    dot("$L$", L, dir(270));
    dot("$E$", E, dir(100));
    dot("$H$", H, dir(45));

    /* Source generated by TSQ */
  \end{asy}
\end{center}

The main idea is a ``floating orthocenter'' amidst a large number of reflections.
This motivates us to reflect $H$ over $\ol{BC}$ to $A'$
and $H$ over $\ol{AB}$ to $C'$.
Then the problem is equivalent to trying to prove that
the points $E$, $M$, $A'$ are collinear.

Let $E'$ be the second intersection of line $A'M$ with $\Gamma$.
First, we claim that $E'$, $C'$, $L$ are collinear.
Note that and \[ \angle LC'B = \angle KHB = \angle MA'B = \angle E'A'B \] (using the reflections).
From here we get $\angle LC'B + \angle BC'E' = 180^{\circ}$, and
hence the collinearity is proved.
On the other hand,
\[ \angle LBM =
  2\left( \angle KBA - KBC \right)
  = 2 \angle B \]
and
\[ \angle C'E'A' = 180 -2 \angle B. \]
So $L$, $B$, $M$, $E'$ are cyclic as desired.