Question

In a traingle ABC IF AD perpendicular BC and AD^2=BD×DC then prove that

Answer 1

Given that AD2=BD⋅DCAD2=BD⋅DC, then ADDC=BDADADDC=BDAD.

Let’s see the triangles ABDABD and ACDACD. They are both right triangles. And we have that tan∠BAD=BDADtan⁡∠BAD=BDAD, and tan∠ACD=ADDCtan⁡∠ACD=ADDC, therefore ∠BAD=∠ACD∠BAD=∠ACD. (1)

As ∠ADB=∠CDA=90∘∠ADB=∠CDA=90∘, then triangles ABDABD and CADCAD are similar. This means that ∠CAD=∠ABD∠CAD=∠ABD, where ∠CAD+∠ADC+∠DCA=180∘∠CAD+∠ADC+∠DCA=180∘, so ∠CAD+∠ACD=90∘∠CAD+∠ACD=90∘. (2)

Applying (1) in (2), ∠CAD+∠BAD=90∘∠CAD+∠BAD=90∘, therefore ∠BAC=90∘∠BAC=90∘ and triangle ABCABC is right with AA a right angle.