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Intersection of Angle Bisectors of Triangles


Date: 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
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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