SOLUTION: In any triangle ABC, E is any point on altitude line segment AD. Prove that (AC)^2-(CE)^2= (AB)^2-(EB)^2

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Question 1061271: In any triangle ABC, E is any point on altitude line segment AD. Prove that (AC)^2-(CE)^2= (AB)^2-(EB)^2
Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
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In any triangle ABC, E is any point on altitude line segment AD. Prove that (AC)^2-(CE)^2= (AB)^2-(EB)^2
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0.  Make a sketch to follow my arguments.


1.  From the triangle ADC:  abs%28DC%29%5E2 = abs%28AC%29%5E2+-+abs%28AD%29%5E2.

    From the triangle EDC:  abs%28DC%29%5E2 = abs%28CE%29%5E2+-+abs%28ED%29%5E2.

    It implies  abs%28AC%29%5E2+-+abs%28AD%29%5E2 = abs%28CE%29%5E2+-+abs%28ED%29%5E2.

    Hence,      abs%28AC%29%5E2+-+abs%28CE%29%5E2 = abs%28AD%29%5E2+-+abs%28ED%29%5E2.   (1)


2.  Similarly,
    From the triangle ADB:  abs%28BD%29%5E2 = abs%28AB%29%5E2+-+abs%28AD%29%5E2.

    From the triangle EDC:  abs%28BD%29%5E2 = abs%28BE%29%5E2+-+abs%28ED%29%5E2.

    It implies  abs%28AB%29%5E2+-+abs%28AD%29%5E2 = abs%28BE%29%5E2+-+abs%28ED%29%5E2.

    Hence,      abs%28AB%29%5E2+-+abs%28BE%29%5E2 = abs%28AD%29%5E2+-+abs%28ED%29%5E2.   (2)


3.  From (1) and (2) you have 

    abs%28AB%29%5E2+-+abs%28BE%29%5E2 = abs%28AC%29%5E2+-+abs%28CE%29%5E2.


It is what has to be proved.

Solved.