Question

If $BM$ and $CN$ are the perpendicular drawn on the sides $AC$ and $AB$ of the triangle $ABC$, prove that the points $B,C,M$ and $N$ are concyclic.

Answer 1

Prove that the points are concyclic

According to the given data

$BM$ and $CN$ are the perpendicular drawn on the sides $AC$ and $AB$ of the triangle $ABC$

We have to prove that the points $B,C,M$ and $N$ are concyclic.

$BM$ and $CN$ are the perpendiculars drawn on the sides $AC$ and $AB$of the triangle

So, we have,

$\angle BMC=\angle BNC=90°$

We know that, if a line segment joining two points subtends equal angles on the same side of the line containing the segment, then the four points are concyclic.

As per the given data

Since $BC$ joins the two points, $B$ and $C$, subtending equal angles $\angle BMC$ and $\angle BNC$ at $M$ and $N$ on the same side $BC$ containing the segment, then $B,C,M$ and $N$ are concyclic.

Hence, we get that, $B,C,M$ and $N$ are concyclic.