Triangle A B C is shown. Lines are drawn from each point to the opposite side and intersect at point D. They form line segments A G, B E, and C F.

Question

Triangle A B C is shown. Lines are drawn from each point to the opposite side and intersect at point D. They form line segments A G, B E, and C F.

In the diagram, which must be true for point D to be an orthocenter?

BE, CF, and AG are angle bisectors.
BE ⊥ AC, AG ⊥ BC, and CF ⊥ AB.
BE bisects AC, CF bisects AB, and AG bisects BC.
BE is a perpendicular bisector of AC, CF is a perpendicular bisector of AB, and AG is a perpendicular bisector of BC.

Answer 1

Bot GPT 4-o mini answered
1 hour ago

For point D to be the orthocenter of triangle ABC, it is necessary that the lines drawn from each vertex (A, B, and C) to the opposite side are altitudes. This means that each of these lines is perpendicular to the side it intersects.

Therefore, for D to be the orthocenter, the correct statement is:

BE ⊥ AC, AG ⊥ BC, and CF ⊥ AB.

This indicates that line BE is perpendicular to side AC, line AG is perpendicular to side BC, and line CF is perpendicular to side AB. In this configuration, point D would indeed be the orthocenter of triangle ABC.