proof of triangle incenter
In and construct bisectors of the angles at and , intersecting at 11Note that the angle bisectors must intersect by Euclid’s Postulate 5, which states that “if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” They must meet inside the triangle by considering which side of and they fall on. Draw . We show that bisects the angle at , and that is in fact the incenter of .
Drop perpendiculars from to each of the three sides, intersecting the sides in , , and . Clearly, by AAS, and also . Thus . It follows that is the incenter of since its distance from all three sides is equal.
Also, since we see that and are right triangles with two equal sides, so by SSA (which is applicable for right triangles), . Thus bisects .
|Title||proof of triangle incenter|
|Date of creation||2013-03-22 17:12:26|
|Last modified on||2013-03-22 17:12:26|
|Last modified by||rm50 (10146)|