# Similar Trianles,pythagoras

• Mar 6th 2007, 06:12 AM
Riny
Similar Trianles,pythagoras
In An Equilateral Triangle -'abc', 'd' Is A Point On 'bc' Such That Bd=1/3bc .prove That 9ad^2 =7 Ab^2
• Mar 6th 2007, 07:52 AM
Soroban
Hello, Riny!

I have a proof, but I don't know if it satisfies your needs . . .

Quote:

In equilateral triangle ABC, D is a point on BC such that BD = (1/3)BC.

Code:

```                A                 *               *:*               * : *             *  :  *             *  :  *           *    :    *         6 *    :    * 6         *      :      *         *      :      *       *        :        *       *        :        *     *          :E    3    *     *-------*---*-----------*     B - 2 - D - - - 4 - - - C```

Let the side of the triangle be 6: .AB = BC = CA = 6
. . Then: .BD = 2, .DC = 4

Draw altitude AE to side BC.
. . Then: .BE = EC = 3 and DE = 1

Triangle AEB is a 30-60 right triangle.
- - Hence: -AE = 3√3

In right triangle AED: .ADČ .= .DEČ + AEČ .= .1Č + (3√3)Č .= .28

And we have: .9·ADČ .= .9·28 .= .252
. . . - - - and: .7·ABČ .= .7·6Č .= .252