AB = BC and AD is perpendicular to CD. Prove that AC2 = 2.BC. DC

Question:In the figure AB = BC and AD is perpendicular to CD. Prove that AC2 = 2.BC. DC

solution :In &#x2206 ADC
AC² = AD² + DC²
AC² = AB² – BD² + DC²(AS ABD is a right angled triangle)
or AC² = AB² + DC² – BD²
AC² = AB² + DC² – ( DC-BC )²
or AC² = AB² + DC² – (DC² + BC ² -2 DC. BC)
or AC² = AB² + DC² – DC² – BC ² + 2 DC. BC
or AC² = AB² – AB ² + 2 DC. BC (As AB = BC )
.’. AC² = 2 DC. BC Proved