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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