Does the orthocenter of an obtuse triangle lie at a vertex?

Question

I believe it does, but I'm not sure.

Answer 1

No, it lies outside the triangle.

The orthocenter of a right triangle lies at the vertex where the two legs of the triangle meet.

Answer 2

i do not think it is one of the vertex, however it may lie on the line that is perpendicular to the vertex. according to the definition orthocenter is the point where the perpendiculars from the three vertices meet. for an obtuse triangle, two of these perpendiculars will lie on the extended opposite sides and the third one including the obtuse angle will lie on the third side of the triangle. so this third perpendicular needs to be extended to obtain the intersection of the other two perpendiculars.

Answer 3

If one angle is obtuse, then the other two angles are acute.

From the two vertices at which the angles are acute, the perpendiculars dropped will meet the opposite sides only if that side is extended.

Therefore the orthocenter is outside and not at the vertex.

Answer 4

You cant better MathOpenRef for geometrical constructions and interactive animations: For the othocentre yuo have to get altitudes, which pass through opposite vertices of the triangle Let ABC be trg, altidude of AB passes thru vertex C altitude of BC passes thru vertex A where altitudes intersect is the orthocentre. Construction is animated using compass and straight-edge in second link below: