Question

in triangle ABC ,AB=AC and BD is perpendicular to AC prove that BD square+CDsquare=2AC.CD

Answer 1

Given: AB=AC , BD _|_ AC

To Prove: BD2 -DC2 = 2DC* AD

Proof: BC2 =CD2 + BD2 [PYTHA. THEO.]

=> BD2 =BC2-CD2

AB2 = AD2+ BD2 [PYTHA. THEO.]

=> BD2 =AB2-AD2

=[ AB+AD ] * [ AB-AD ] {A2-B2 = [A+B] [A-B] }

=[ AC+AD ] [ AC-AD] { Since AB=AC}

=[ AC+AD ] * DC

=AC*DC + AD*DC

=[AD+DC]*DC + AD*DC

=AD*DC+DC2+AD*DC

BD2 =2AD*DC+DC2

BD2 -DC2 = 2DC* AD