In an equilateral ABC, D is a point on side BC such that  Prove that 9AD2 = 7AB2.  - Zigya
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In an equilateral ABC, D is a point on side BC such that BD equals 1 third BC. Prove that 9AD2 = 7AB2. 



Given : An equilateral triangle ABC such that To Prove : 9AD2 = 7AB

Given : An equilateral triangle ABC such that BD equals 1 third BC.
To Prove : 9AD2 = 7AB2
Const: Draw AE ⊥ BC.
Proof: In right triangles ABE and ACE, we have
AE = AE    [common]
∠AEB = ∠AEC    [90°]
and    AB = AC
[∆ABC is an equilateral]
Therefore, by using RHS congruent condition
                                       space space increment ABE approximately equal to increment ACE
rightwards double arrow                                          BE = CE         [by CPCT]
rightwards double arrow                                          BE equals CE equals 1 half BC
In right triangle ADE, we have
                          space AD squared space equals space AE squared plus DE squared
rightwards double arrow                     AD squared space equals space AE squared plus left parenthesis BE minus BD right parenthesis squared
rightwards double arrow                  space space space AD squared equals AE squared plus BE squared plus BD squared minus 2 BD. BE
rightwards double arrow                     AD squared space equals space left parenthesis AE squared plus BE squared right parenthesis plus BD squared minus 2 BD. BE
                            open square brackets because space space BD equals 1 third BC space and space BE space equals space CE space equals space 1 half BC close square brackets
rightwards double arrow space space space AD squared space equals space AB squared plus open parentheses 1 third BC close parentheses squared minus 2 cross times 1 third BC cross times 1 half BC
rightwards double arrow space space space AD squared space equals space AB squared plus 1 over 9 BC squared minus BC squared over 3

rightwards double arrow  AD squared equals fraction numerator 9 AB squared plus BC squared minus 3 BC squared over denominator 9 end fraction
rightwards double arrow             9 AD squared space equals space 9 AB squared minus 2 BC squared
But             AB = BC = CA
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