In parallelogram ABCD, Altitudes AF and DE is drawn on DC and produced BA.
In ΔDEA, by pythagoras theorem, DE² + EA² = DA² = …(1)
In ΔDEB. by pythagoras theorem, DE² + EB² = DB²
⇒ DE² + (EA + AB)² = DB²
⇒ (DE² + EA²) + AB² + 2EA × AB = DB²
⇒ DA² + AB² + 2EA × AB = DB² …(ii)
In ΔADF, by Pythagoras theorem, AD² = AF² + FD²
In ΔAFC, by pythagoras theorem
AC² = AF² + FC² = AF² + (DC – FD)² = AF² + DC² + FD² – 2DC × FD
= (AF² + FD²) + DC² – 2DC × FD
⇒ AC² = AD² + DC² – 2DC × FD …(iii)
ABCD is a parallelogram.
Therefore
AB = CD …(iv)
and, BC = AD …(v)
In ΔDEA and ΔADF,
∠DEA = ∠AFD [Each 90]
∠EAD = ∠ADF [EA II DF]
AD = AD [Common]
∴ΔEAD ≅ ΔFDA [AAS congruency rule]
⇒ EA = DF …(vi)
Adding equations (ii) and (iii), we have
DA² + AB² + 2EA AB + AD² + DC² – 2DC FD = DB² + AC²
⇒ DA² + AB² + AD² + DC² + 2EA × AB – 2DC × FD = DB² + AC²
⇒ BC² + AB² + AD² + DC² + 2EA × AB – 2AB × EA = DB² + AC² [From the equation (iv) and (vi)]
⇒ AB² + BC² + CD² + DA² = AC² + BD²