Question

If BM and CN are the perpendiculars drawn on the sides AB and AC of ABC, then prove that the points B,C, M and N are cyclic.

Answer 1

As angle BNC = 90° (given that CN perpendicular to AB of triangle ABC)

So, we can say that points B, N, C lie on a semi circle, the diameter of which is BC. ( as angle on a semicircle= 90°)

Now, angle BMC= 90° ( given that BM perpendicular to AC of triangle ABC

So, points B, M, C too lie on a semicircle the diameter of which is the same BC.

So, diameter is the same for both the semicircles.

That shows that B,C, M, N are lying on the same semi circular arc. So these points are concyclic.

Hence Proved

Answer 2

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