Bot GPT 4-o mini Given that point H is the circumcenter of triangle DEF, we know the following properties about the circumcenter and its relationship to the triangle and the circle that passes through its vertices:
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Point H is the center of the circle that passes through points D, E, and F.
- This statement is true because by definition, the circumcenter of a triangle is the point that is equidistant from all three vertices, making it the center of the circumcircle (the circle that passes through the vertices of the triangle).
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Point H is the center of the circle that passes through points L, M, and N.
- This statement is not necessarily true. Points L, M, and N are the feet of the perpendiculars dropped from H to the sides of triangle DEF, and there is no given information that suggests they are equidistant from H.
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Line segment HE is congruent to line segment HD.
- This statement is true because both segments are radii of the circumcircle of triangle DEF. Since H is equidistant from the vertices D, E, and F, HE = HD.
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Line segment LH is congruent to line segment NH.
- This statement is not necessarily true. L and N are the feet of the perpendiculars from H to the sides of the triangle, and we cannot infer that they are equidistant from H.
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Line segment FL is congruent to line segment FN.
- This statement is also not necessarily true. Similar to the previous options, L and N are specific points on different lines, and there is no property guaranteeing their distances to F are equal.
Therefore, the two statements that must be true are:
- Point H is the center of the circle that passes through points D, E, and F.
- Line segment HE is congruent to line segment HD.
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