The Importance of the Circumcenter in a Triangle | Properties and Theorems

Point H is the circumcenter of triangle DEF.Which must be true? Check all that apply.

In a triangle, the circumcenter is the point of intersection of the perpendicular bisectors of its sides

In a triangle, the circumcenter is the point of intersection of the perpendicular bisectors of its sides. Therefore, the following statements must be true if point H is the circumcenter of triangle DEF:

1. The perpendicular bisectors of sides DE, DF, and EF pass through point H: The circumcenter is equidistant from the vertices of the triangle, so the perpendicular bisectors of the sides must intersect at point H.

2. The distances from the circumcenter H to the vertices D, E, and F are equal: The circumcenter is equidistant from the vertices of the triangle, so the distances HD, HE, and HF must be equal.

3. Angle DHE = Angle DFE = Angle DEF: The circumcenter is the center of the triangle’s circumcircle, which is the circle passing through all three vertices of the triangle. Therefore, the angles subtended by the chords DE, DF, and EF on the circumference of the circumcircle are equal.

In summary, all three of these statements must be true if point H is the circumcenter of triangle DEF.

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