a and b are two sides adjacent to the right angle vertex B of a right angle ΔABC and p is perpendicular drawn from the B on side AC.  Then p² is equal to:

\(\frac{a^2 b^2}{a^2 + b^2}\)

Hypotenuse (AC) = \(\sqrt {AB^2 + BC^2}\) = \(\sqrt {a^2 + b^2}\)

Area of Δ ABC = \(\frac{1}{2}\) AB × BC or \(\frac{1}{2}\) AC × BD 

⇒ \(\frac{1}{2}\) AB × BC = \(\frac{1}{2}\) AC × BD

⇒ AB × BC = AC × BD 

⇒ ab = \(\sqrt {a^2 + b^2}\) × p

squaring both sides

⇒ a² b² = (a² + b²) × p² 

⇒ p²  = \(\frac{a^2 b^2}{a^2 + b^2}\)