Question

Please answer the following
#7

Answer 1

For quick and easy solution, you can do this way... 

From the answers options, see that the answer does not depend on the type of the triangle... So it is true for all types of triangle.    

So assume an equilateral triangle ABC.  O is centroid.

AB= BC = CA = a
AO= BO = CO = 2/3 * altitude = 2/3 * (√3 a /2) = a /√3

AO² + BO² + CO² = a² 

AB² + BC² + CA² = 3 * (AO² + BO²+ CO²)

Answer is = (4)...
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If we want to do for a general triangle:  Use trigonometry..  This is proved by Apollonius. & his theorem.

In triangle: ADC:   AC² = AD² + CD² - 2 AD * CD * Cos(ADC)
In triangle  ADB:    AB² = AD² + BD² - 2 AD * BD * Cos(ADB)
                                     = AD² + BD² -2 AD *BD * [ - Cos(ADC) ]

Adding the two:
    AB² + AC² = 2 (AD² + CD²) = 2 [ (3/2 *OA)² + BC²/4 ]
                      = 9/2  OA² + BC² /2
Similarly,   AC² + BC² = 9/2 OC² + AB² /2
                 AB² + BC² = 9/2  OB² + AC² /2

Add them:  
2(AC² + AB² + BC² )
   = 9/2 (OA² + OB² + OC²)  + (AB²  + BC² + CA²) /2 
 
So     AB² + BC² + CA² = 3 (OA² + OB² + OC²)

Answer 2

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