Question

point h is the circumcenter of triangle def. which must be true? select two options
□ point h is the center of the circle that passes through points d, e, and f.
□ point h is the center of the circle that passes through points l, m, and n.
□ $overline{he} \cong \overline{hd}$
□ $overline{lh} \cong \overline{nh}$
□ $overline{fl} \cong \overline{fn}$

Explanation:

1. Recall the definition of the circumcenter of a triangle: The circumcenter is the center of the circumscribed circle (circumcircle) that passes through all three vertices of the triangle. So for triangle \( DEF \), the circumcenter \( H \) is the center of the circle passing through \( D \), \( E \), and \( F \). This makes the first option true.
2. Analyze the property of the circumcenter: The circumcenter is equidistant from all three vertices of the triangle. So \( HD = HE = HF \) (since \( H \) is the circumcenter of \( \triangle DEF \)), which means \( \overline{HE} \cong \overline{HD} \). This makes the third option true.
3. For the second option: There is no information given that \( L \), \( M \), \( N \) are related to the circumcircle of \( \triangle DEF \), so we cannot conclude that \( H \) is the center of the circle through \( L \), \( M \), \( N \).
4. For the fourth option: \( LH \) and \( NH \) are not necessarily congruent. The perpendicular bisectors' segments from \( H \) to the sides (like \( LH \), \( NH \), \( MH \)) are not guaranteed to be equal unless the triangle has specific properties (e.g., isoceles or equilateral), which is not indicated here.
5. For the fifth option: \( FL \) and \( FN \): \( L \) and \( N \) are on different sides (\( FD \) and \( FE \) respectively), and there's no reason for \( FL \) and \( FN \) to be congruent. \( L \) is the foot of the perpendicular from \( H \) to \( FD \), and \( N \) is the foot of the perpendicular from \( H \) to \( FE \), but unless \( FD = FE \), \( FL

eq FN \).

Answer:

A. Point H is the center of the circle that passes through points D, E, and F
C. \(\overline{HE} \cong \overline{HD}\)