in triangle ABC,if AD is perpendicular to BCand AD*2=BD.DC,prove that angle BAC=90 degree

 Given : In triangle ABC , AD is perpendicular to BC & AD2 = BD.DC

To proove : BAC = 90 degree

Proof : in right triangles ADB & ADC we have AB2 = AD2 +BD2 . . . . . . . . .1

& AC2 = AD2+ DC2 . . . . . . . .2

from 1 & 2 , AB2 + AC2 = 2AD2 . BD2 + DC2      

                                          = 2BD . CD + BD2 + CD2         { AD*2 = BD . CD (GIVEN ) }

                                          = (BD + CD )2 = BC2

Thus in triangle ABC we have , AB2 + AC2 = BC2

hence triangle ABC is a right triangle right angled at  A

so BAC = 90 degree      ( proved )

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