Intersection of Angle Bisectors of TrianglesDate: 02/17/98 at 11:38:43 From: Jon Swaine Subject: bisectors of each angle of triangle I know that bisectors of each angle of a triangle intersect at one point because it works for every triangle I try. But how can I generalize this for a proof? Thanks for your time. Jon Swaine Date: 02/17/98 at 12:18:25 From: Doctor Wilkinson Subject: Re: bisectors of each angle of triangle Let me give you the general idea and leave it to you to work out the details. Let the triangle be ABC. Let the bisector of the angle at A and the bisector of the angle at B intersect at point X. We're going to show that XC bisects the angle at C. We're going to find three right triangles. Let XP be perpendicular to AB, XQ perpendicualr to AC, and XR perpendicular to BC. Now you have three pairs of right triangles: XAP and XAQ XBP and XBR XCQ and XCR Now, XAP and XAQ are right triangles that share the side AX, and the angles QAX and PAX are equal, because AX is the bisector of the angle at A. From this you can show that XAP and XAQ are congruent, and therefore that XQ and XP are congruent. Similarly, you can show that XP and XR are congruent, and therefore that XQ and XR are congruent. From this you can conclude that the triangles XCQ and XCR are congruent; and you should be able to wrap it up from there. -Doctor Wilkinson, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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