All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Triangle's bisector theorem (Posted on 2015-11-07)
Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D between B and C.
The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC.
BD/DC= AB/AC
Prove it.

 No Solution Yet Submitted by Ady TZIDON No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 A Third Solution Comment 2 of 2 |
`VECTOR SOLUTION Two uppercase letters such as PQ will denote avector from point P to point Q. Its length willbe denoted by |PQ|. Single lowercase letterswill represent real numbers. In our problem wehave a = |BC|, b = |AC|, and c = |AB|.`
`   AD = t*( AB/c + AC/b )         for some t`
`   AD = w*AB + (1-w)*AC           for some w`
`Combining these two equations gives`
`   AD = ( b*AB + c*AC )/(b+c)`
`-------------------------------------------`
`   BD = AD - AB = c*( AC - AB )/(b+c) = c*BC/(b+c)`
`   |BD| = a*c/(b+c)`
`   DC = AC - AD = b*( AC - AB )/(b+c) = b*BC/(b+c)`
`   |BD| = a*b/(b+c)`
`Therefore,`
`   |BD|/|DC| = c/b = |AB|/|AC|`
`QED`

Edited on November 9, 2015, 7:46 pm
 Posted by Bractals on 2015-11-09 13:25:56

 Search: Search body:
Forums (0)