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May 8th, 2015, 05:21 AM  #1 
Senior Member Joined: Jan 2015 From: USA Posts: 107 Thanks: 2  Bisector and inner circle. Hard to solve
On the plane xy there are four points: O(0, 0) A(0, 3) B(0, 3), C(4,0). When point D is the intersections of bisector of angle ABC (Angle B)and the xaxis, then OD : DC = ? : ?. and the coordinates of the inner center of ABC are (?/2; ?) I have nooo idea how to solve it I would appreciate your help 
May 8th, 2015, 06:53 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
I'm not sure this is the simplest way to do it but a "brute strength" method would be to use the cosine law to find cosine of angle B: The side opposite angle B has length 5 and the other two sides are of length 5 and 6: $b^2= a^2+ c^2 2ac cos(B)$ 25= 25+ 36 2(30)cos(B) co(B)= 36/60= 6/10= 0.6 The bisector makes angle B/2 with the yaxis and $cos(B/2)= \sqrt{\frac{cos(B)+ 1}{2}}$. Here that is $cos(B/2)= \sqrt{0.8}$ so $sin(B/2)= \sqrt{1 0.8}= \sqrt{0.2}$ and $tan(B/2)= \frac{\sqrt{.2}}{\sqrt{.8}}= \sqrt{\frac{1}{4}}= \frac{1}{2}$. Since $tan(B/2)= \frac{OD}{OB}$, $\frac{1}{2}= \frac{OD}{3}$ so OD= 3/2. Last edited by Country Boy; May 8th, 2015 at 06:59 AM. 
May 8th, 2015, 07:44 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1123 Math Focus: Elementary mathematics and beyond 
Since the intersection of the angle bisectors of a triangle is the center of its incircle, OD = r and DC = 4  r, where r is the inradius. To compute the inradius use A = Sr, where A is the area of the triangle and S is the semiperimeter (half the perimeter). 

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