Question

If  p₁, p₂, p₃ are the altitudes of a triangle from the vertices A, B, C and ∆ the area of the triangle, then prove that 1/p₁ + 1/p₂ - 1/p₃ = 2ab/(a + b + c)∆cos²(C/2)

Answer 1

Since,  ∆ = 1/2ap₁ ⇒ 1/p₁ = a/2∆

Similarly, 1/p₂ = b/2∆, 1/p₃ = c/2∆

∴ 1/p₁ + 1/p₂ - 1/p₃ = 1/2∆(a + b - c)

= (2(s - c))/2∆ = (s - c)/∆ = s(s - c)/ab x ab/s∆

= ab/((a + b + c)/2)∆ x cos²(C/2) = (2ab/(a + b + c)∆)cos²(C/2)