In a $\triangle ABC, E$ and $F$ are the mid-points of $AC$ and $AB$ respectively. The altitude $AP$ to $BC$ intersects $FE$ at $Q$. Prove that $AQ = QP$.
Given:
In a $\triangle ABC, E$ and $F$ are the mid-points of $AC$ and $AB$ respectively. The altitude $AP$ to $BC$ intersects $FE$ at $Q$.
To do:
We have to prove that $AQ = QP$.
Solution:
Join $EF$.
$AP \perp BC$ is drawn which intersects $EF$ at $Q$ and meets $BC$ at $P$.
![](/assets/questions/media/153848-53340-1632932271.png)
In $\triangle ABC$,
$EF \parallel BC$ and $EF = \frac{1}{2}BC$
$\angle F = \angle B$
In $\triangle ABP$,
$F$ is the mid point of $AB$ and $Q$ is the mid point of $FE$
This implies,
$FQ \parallel BC$
$Q$ is the mid point of $AP$.
$AQ = QP$
Hence proved.
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