Difference between revisions of "Angle Bisector Theorem"

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#REDIRECT[[Angle bisector theorem]]
 
 
== Introduction ==
 
The '''Angle Bisector Theorem''' states that given [[triangle]] <math>\triangle ABC</math> and [[angle bisector]] AD, where D is on side BC,  then <math> \frac cm = \frac bn </math>. It follows that <math> \frac cb = \frac mn </math>. Likewise, the [[converse]] of this theorem holds as well.
 
 
 
 
 
Further by combining with [[Stewart's Theorem]] it can be shown that <math>AD^2 = b\cdot c - m \cdot n</math>
 
 
 
<asy> size(200); defaultpen(fontsize(12)); real a,b,c,d; pair A=(1,9), B=(-11,0), C=(4,0), D; b = abs(C-A); c = abs(B-A); D = (b*B+c*C)/(b+c); draw(A--B--C--A--D,black); MA(B,A,D,2,green); MA(D,A,C,2,green); label("$A$",A,(1,1));label("$B$",B,(-1,-1));label("$C$",C,(1,-1));label("$D$",D,(0,-1)); dot(A^^B^^C^^D,blue);label("$b$",(A+C)/2,(1,0));label("$c$",(A+B)/2,(0,1));label("$m$",(B+D)/2,(0,-1));label("$n$",(D+C)/2,(0,-1)); </asy>
 
 
 
== Proof ==
 
 
 
By LoS on ACD and ABD,
 
 
 
AB/BD=sin(BDA)/sin(BAD) ... (1) and
 
AC/AD=sin(ADC)/sin(DAC) ... (2)
 
Well, we also know that BDA and ADC add to 180 degrees. I think that means that we can use sin(180-x)=sin(x) here. Doing so, we see that
 
sin(BDA)=sin(ADC)
 
I noticed that these are the numerators of (1) and (2) respectively. Since BAD and DAC are equal, then you get the equation for the bisector angle theorem. ~ sachi2019
 
 
 
== Examples ==
 
 
 
# Let ABC be a triangle with angle bisector AD with D on line segment BC.  If <math> BD = 2, CD = 5,</math> and <math> AB + AC = 10 </math>, find AB and AC.<br> '''''Solution:''''' By the angle bisector theorem, <math> \frac{AB}2 = \frac{AC}5</math> or <math> AB = \frac 25 AC </math>.  Plugging this into <math> AB + AC = 10 </math> and solving for AC gives <math> AC = \frac{50}7</math>.  We can plug this back in to find <math> AB = \frac{20}7 </math>.
 
# In triangle ABC, let P be a point on BC and let <math> AB = 20, AC = 10, BP = \frac{20\sqrt{3}}3, CP = \frac{10\sqrt{3}}3 </math>.  Find the value of <math> m\angle BAP - m\angle CAP </math>. <br> '''''Solution:'''''  First, we notice that <math> \frac{AB}{BP}=\frac{AC}{CP} </math>.  Thus, AP is the angle bisector of angle A, making our answer 0.
 
# Part '''(b)''', [[1959 IMO Problems/Problem 5]].
 
 
 
== See also ==
 
* [[Angle bisector]]
 
* [[Geometry]]
 
* [[Stewart's Theorem]]
 
 
 
[[Category:Geometry]]
 
 
 
[[Category:Theorems]]
 

Latest revision as of 16:23, 9 May 2021

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