Pedal triangle of a Triangle: Let Δ ABC be any triangle and let D, E, F be the feet of perpendiculars from the vertices A, B, C on the opposite sides BC, CA, AB respectively, then the triangle DEF is known as pedal triangle of Δ ABC. H is the orthocentre of the Δ ABC. We note that ∠ HDC = ∠ HEC = 90º, so the points H, D, C and E are concyclic.

∴ ∠ HDE = ∠ HCE = 90º – A

Similarly ∠ HDF = ∠ HBF = 90º – A

Hence ∠ FDE = 180º – 2A

Identically we can find

∠ EFD = 180º – 2C

Thus, the angles of pedal triangle are 180º – 2A, 180º –

2B, 180º – 2C.

Further in Δ BFD.

∠ FDB = 90º – (90º – A) = A

∴ FD = b cos B similarly EF = a cos A and DE = c cos C. Thus the sides of pedal triangle are a cos A, b cos B and c cos C. (or R sin2A, R sin2B, R sin2C), where R is the circum-radius of Δ ABC.

The triangle ABC is assumed an acute angled triangle. In case the triangle. ABC be obtuse angled with A as obtuse angle, then the angles of pedal triangle will be represented by

2A – 180º, 2B, 2C and the sides will be represented by

a cos A , b cos B, c cos C.

On the basis of above passage, answer the following questions:

(i) If λ , m, n denote the sides of a pedal triangle Then

=