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 C
F ⊥ 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.
2 answers:
BE, CF and AG are angle bisectors. Therefore, their meeting point is I, the incentre of the triangle ABC.
BE ⊥ AC, AG ⊥ BC, and CF ⊥ AB. Therefore, meeting point of BE, CF and AG is O, the orthocentre of the triangle ABC.
BE bisects AC, CF bisects AB, and AG bisects BC. Therefore, BE, CF and AG are medians and their meeting point is G, the centroid of the triangle ABC.
From the above, it is clear that I = O = G.
This property holds good only for equilateral triangles.
Hence, Δ ABC is an equilateral triangle.
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