The third side of triangle is perpendicular to one of the bisectors of the angles between two sides. The equation of bisectors are ‌
|3−4y−2|
√32+(−4)2
=‌
|12x−5y+6|
√122+(−5)2
‌⇒‌
3x−4y−2
5
=±‌
(12x−5y+6)
13
‌⇒±(39x−52y−26)=(60x−25y+30) ‌⇒(60±39)x−(25±52)y−(30∓26)=0 ‌⇒99x−77y−4=0‌ and ‌21x+27y−56=0 Slope of the bisectors are ‌
9
7
and ‌
7
9
. Slopes of the third side is −‌
7
9
and −‌
9
7
. ‌y−y1=−‌
7
9
(x−x1)‌ or ‌(y−y1)=−‌
9
7
(x−x1) ‌⇒‌‌9y−9y1=−7x+7x1 or 7y−7y1=−9x+9x1 ‌7x+9y−9y1−7x1=0 or 9x+7y−7y1−9x1=0 ‌9x+7y+C=0‌‌[∴C=−7y1−9x1]