Question

in the diagram, which must be true for point d to be an orthocenter?

• $overline{be}$, $overline{cf}$, and $overline{ag}$ are angle bisectors.
• $overline{be} \perp overline{ac}$, $overline{ag} \perp overline{bc}$, and $overline{cf} \perp overline{ab}$.
• $overline{be}$ bisects $overline{ac}$, $overline{cf}$ bisects $overline{ab}$, and $overline{ag}$ bisects $overline{bc}$.
• $overline{be}$ is a perpendicular bisector of $overline{ac}$, $overline{cf}$ is a perpendicular bisector of $overline{ab}$, and $overline{ag}$ is a perpendicular bisector of $overline{bc}$.

Explanation:

The orthocenter of a triangle is the intersection point of its three altitudes. An altitude is a line segment from a vertex perpendicular to the opposite side. For point D to be the orthocenter, the lines through the vertices (BE, CF, AG) must be altitudes—i.e., each must be perpendicular to the opposite side.

Answer:

BE ⊥ AC, AG ⊥ BC, and CF ⊥ AB.