If $P, Q$ and $R$ are the mid-points of the sides, $B C, C A$ and $A B$ of a triangle and $A D$ is the perpendicular from $A$ on $B C$, then prove that $P, Q, R$ and $D$ are concyclic.
Given in $\triangle A B C, P, Q$ and $R$ are the mid-points of the sides $B C, C A$ and $A B$ respectively Also, $A D \perp B C$. .
To prove $P, Q, R$ and $D$ are concyclic.
Construction Join $D R, R Q$ and $Q P$
Proof in right angled $\triangle A D P, R$ is the mid-point of $A B$.
$\therefore \quad R B=R D$
$\Rightarrow$ $\angle 2=\angle 1$ ...(i)
[angles opposite to the equal sides are equal]
Since $B$ and $Q$ are the mid-points of $A B$ and $A C$, then
$R Q \| B C$ [by mid-point theorem]
or $R Q \| B P$
Since, $Q P \| R B$, then quadrilateral $B P Q R$ is a parallelogram.
$\Rightarrow$ $\angle 1=\angle 3$ ...(ii)
[opposite angles of parallelogram are equai]
From Eqs. (i) and (ii), $\angle 2=\angle 3$
But $\angle 2+\angle 4=180^{\circ}$ [linear pair axiom]
$\therefore$ $\angle 3+\angle 4=180^{\circ}$ $[\because \angle 2=\angle 3]$
Hence, quadrilateral $P Q R D$ is a cyclic quadrilateral. So, points $P, Q, R$ and $D$ are con-cyclic.
Hence proved.
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