The incenter of a right triangle is equidistant from the midpoint of the hypotenuse and the vertex of the right angle. Show that the triangle contains a 30 degree angle.
The incenter of a triangle is determined by the intersection of two angle bisectors.For a 30-60-90 tri the bisector of the 60 angle is also the perpendicular bisector of the median drawn to the hypothenus.Therfore the distance from the incenter to the midpoint of hypothenuse and to the vertex of the 90 angle are equal.Is this enough for you to write a formal proof?
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