The circle is inscribed in a triangle determine by the lines 3x + 4y - 5 = 0 , 4x - 3y- 65 = 0 and 7x - 24y + 55 = 0. What is the equation of a circle?

Question

The circle is inscribed in a triangle determine by the lines 3x + 4y - 5 = 0 , 4x - 3y- 65 = 0 and 7x - 24y + 55 = 0. What is the equation of the circle?

Answer 1

(3x+4y-5)/5 = (4x - 3y - 65)/5; x - 7y - 60 = 0. This is equation of bisector of angle between first 2 sides.

(4x - 3y - 65)/5 = - (7x - 24y + 55)/25; 20x - 15y - 325 = - 7x+24y - 55; 27x -39y - 270 = 0 this is equation of

bisector between 2nd and s3rd sides. Intersection of this 2 bisectors give us center of inscribed circle.

x - 7y = 60

27x - 39y = 270

x = 60 + 7y, 1620 + 189y - 39y = 270; 150y = - 1350; y = - 5; x = 25; (25, - 5) is center of circle.

Radius of circle is distance from center to any side: r = |3·25 + 4(- 5) - 65|/5 = 2

Equation of inscribed circle: (x - 25)² + (y - 5)² = 4