1. Identify the given elements and their properties:
- Let $Y$ and $Z$ be the feet of the altitudes from $B$ and $C$ respectively in triangle $ABC$.
- Let $U$ and $V$ be the feet of the perpendiculars from $Y$ and $Z$ to the line $BC$.
- The lines $YV$ and $ZU$ intersect at point $L$.

2. **Establish the orthogonality of $AL$ and $BC$:**
- We need to prove that $AL \perp BC$.

3. Use properties of the orthocenter and cyclic quadrilaterals:
- Note that $Y$ and $Z$ are the feet of the altitudes, so $AY \perp BC$ and $AZ \perp BC$.
- Since $U$ and $V$ are the feet of the perpendiculars from $Y$ and $Z$ to $BC$, quadrilaterals $BYUV$ and $CZUV$ are cyclic.

4. Apply the properties of cyclic quadrilaterals:
- In cyclic quadrilateral $BYUV$, the opposite angles are supplementary. Therefore, $\angle BYU + \angle BVU = 180^\circ$.
- Similarly, in cyclic quadrilateral $CZUV$, $\angle CZV + \angle CUV = 180^\circ$.

5. **Analyze the intersection point $L$:**
- Since $YV$ and $ZU$ intersect at $L$, we need to show that $L$ lies on the altitude from $A$ to $BC$.

6. Use the properties of the orthocenter:
- The orthocenter $H$ of triangle $ABC$ is the intersection of the altitudes. Since $Y$ and $Z$ are the feet of the altitudes, $H$ lies on $AY$ and $AZ$.
- The point $L$ is the intersection of $YV$ and $ZU$, which are perpendicular to $BC$ at $U$ and $V$ respectively.

7. Conclude the proof:
- Since $L$ is the intersection of the perpendiculars from $Y$ and $Z$ to $BC$, and these perpendiculars are part of the cyclic quadrilaterals $BYUV$ and $CZUV$, $L$ must lie on the altitude from $A$ to $BC$.
- Therefore, $AL \perp BC$.

$\blacksquare$