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Let $4x -y+2=0, x-4y-8=0, x+4y-8=0$ make a triangle. Find the $(x,y)$ coordinates of the triangle's incenter(center of the inscribed circle).

I know that the intersection of the bisectors make the triangle's incenter. I have also tried to calculate each of the triangle's vertices, but I can't make it any further.

Andrei Onoie
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  • Hint: the gradient of the angle bisector of two lines $L_1, L_2$ is the mean of the gradients of $L_1$ and $L_2$. – Josh Apr 09 '17 at 20:53
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    Hint: the $x$ axis is line bissector of $L_2$ and $L_3$, thus... – Jean Marie Apr 09 '17 at 20:56
  • @JoshChen: so the slope of the line $y=x$ is the average between $0$ and $+\infty$? – Jack D'Aurizio Apr 09 '17 at 20:57
  • Not only two sides are symmetric with respect to the $x$-axis, but the given triangle is also a right triangle. – Jack D'Aurizio Apr 09 '17 at 21:00
  • @Jack D'oh! Replace "gradient" with "angle with the $x$-axis". Of course then the hint becomes not as useful. – Josh Apr 09 '17 at 22:27
  • Note that the normals in the equations of the three lines have equal length, so finding the angle bisectors is very simple. – amd Apr 10 '17 at 05:59

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