The lines might as well be replaced by the sides of an angle. My gut tells me that it must be a segment perpendicular to the angle bisector but I can't justify it.
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If the given point is on the angle bisector, I second your gut. However, think about what happens if the given point is very close to one of the lines. Then the segment would be better off being perpendicular to the opposite leg rather than to the angle bisector. In the light of this, my gut says take the ray from the intersection of lines and through the given point. Reflect this ray about the angle bisector. Maybe it ought to be perpendicular to this ray? – Arthur May 26 '17 at 10:52
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Yeah. I am wrong. If the point is on the side then it would be like comparing the base of an isosceles to perpendicular dropped to the side so it must but not be the segment perpendicular to the angle bisector. – tighten May 26 '17 at 11:17
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Related: https://math.stackexchange.com/q/2297292/265466 – amd May 26 '17 at 20:29
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Actually this post inspired my question. I wanted to approach it using something that does not rely on cartesian system). I want something that is proven using elementary geometry. – tighten May 27 '17 at 03:00