A triangle has corners at #(2 ,3 )#, #(1 ,9 )#, and #(6 ,8 )#. What is the radius of the triangle's inscribed circle?

1 Answer
Cesareo R.
May 14, 2016

#(x-3.0195)²+(y-6.9143)²=1.6491^2#

Explanation:

The triangle vertices #p_1=(2,3),p_2=(1,9),p_3=(6,8)#
First step is bissectrice building.
#b_1->p_1+v_1 lambda_1#
#b_2->p_2+v_2 lambda_2#

where:
#v_1 = (p_2-p_1)/norm(p_2-p_1)+(p_3-p_1)/norm(p_3-p_1) = (0.460296, 1.76726)#
#v_2 = (p_1-p_2)/norm(p_1-p_2)+(p_3-p_2)/norm(p_3-p_2)=(1.14498, -1.18251)#

The bissectrices intersection point is the circumference center calculated as the solution #(lambda_1^0,lambda_2^0)# to the system.
#p_1+v_1 lambda_1 = p_2+v_2 lambda_2#
The circumference center is obtained as
#c=p_1+v_1 lambda_1^0= p_2+v_2 lambda_2^0 = (3.01951, 6.9143)#
The circumference radius is obtained using Pythagoras.
#r = sqrt(norm(c-p_1)^2-norm((c-p_1). (p_2-p_1)/norm(p_2-p_1))^2) = 1.64914#

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