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# How can you prove that the angle bisector of an angle of a triangle must intersect the opposite side?

Question #102155. Asked by Bronxiteone. (Jan 06 09 1:42 PM)

looney_tunes

Using a coordinate geometry approach, identify the coordinates of the vertex of the angle and the equations of the lines forming the two sides; choose an arbitrary point on each line and find the general equation of the line connecting them (the third side of your triangle); write the equation of the line that meets the conditions of angle bisection (that it is equidistant from each of the lines forming the two sides); solve simultaneously the equations for this line and for the third side.

If you are trying to do this as an absolute proof for any angle and triangle, your equations will be full of unknowns (x1, y1, m1, etc), and will need a lot of careful algebraic manipulation. If you have a specific triangle in mind, the presence of numbers makes the solution of the equations much simpler.

http://regentsprep.org/Regents/mathb/1D/Coordinatelesson.htm

Of course, this is not the only method of proof available, but it is the simplest to describe as a general procedure without actually writing out the required proof!

 Jan 06 09, 1:55 PM
zbeckabee

The bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.

That is, for any triangle ABC, the bisector of the angle at C divides the opposite side into segments of length x and y such that a/b = x/y.

More:

http://jwilson.coe.uga.edu/EMT725/Bisect/bisect.html

 Jan 06 09, 2:17 PM
Bronxiteone

Thank you but the proof for this theorem( about dividing the opposite side proportionally) requires the assumption that the line intersects the opposite side.

 Jan 07 09, 5:05 AM
zbeckabee

If it didn't intersect the opposite side...there would be no proof.

 Jan 07 09, 3:22 PM
Baloo55th

If it is the bisector of an internal angle of a triangle, it is by definition internal to the triangle, and by the definition of bisector cannot be the same as either of the adjacent sides. It must therefore intersect the opposite side. I see it as a matter of definition rather than formula or equation.

 Jan 07 09, 3:30 PM
zbeckabee

Bingo!

ANGLE BISECTOR OF A TRIANGLE -- An angle bisector of a triangle is a line segment that extends from a vertex to the side opposite in such a way that it bisects the angle at the vertex.

http://library.thinkquest.org/C004647/to/geom/glossary.html

 Jan 07 09, 5:02 PM

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