[Answer] In which type of triangle is the orthocenter on the perimeter of the triangle?

Answer: A. a right triangle
In which type of triangle is the orthocenter on the perimeter of the triangle?

The three (possibly extended) altitudes intersect in a single point called the orthocenter of the triangle usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. does not have an angle greater than or equal to a right angle). If one angle is a right angle the orthocenter coincides with the vertex at the right angle. Let A B C denote the vertices and also the angles of the triangle and let a = |BC| b = |CA| c = |AB| be the side lengths. The orthocenter has trilinear coordinates

The three (possibly extended) altitudes intersect in a single point called the orthocenter of the triangle usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. does not have an angle greater than or equal to a right angle). If one angle is a right angle the orthocenter coincides with the vertex at the right angle. Let A B C denote the vertices and also the angles of the triangle and let a = |BC| b = |CA| c = |AB| be the side lengths. The orthocenter has trilinear coordinates ${\displaystyle \sec A:\sec B:\sec C=\cos A-\sin B\sin C:\cos B-\sin C\sin A:\cos C-\sin A\sin B }$ and barycentric coordinates ${\displaystyle \displaystyle (a^{2}+b^{2}-c^{2})(a^{2}-b^{2}+c^{2}):(a^{2}+b^{2}-c^{2})(-a^{2}+b^{2}+c^{2}):(a^{2}-b^{2}+c^{2})(-a^{2}+b^{2}+c^{2})}$ ${\displaystyle =\tan A:\tan B:\tan C.}$ Since barycentric coordinates are all positive for a point in a triangle’s interior but at least one is negative for a point in the exterior and two of the barycentric coordinates are zero for a vertex point the barycentric coordinates given for the orthocenter show that the orthocenter is in an acute triangle’s interior on the right-angled vertex of a right triangle and exterior to an obtuse triangle. In the complex plane let the points A B and C represent the numbers ${\displaystyle z_{A}}$ ${\displaystyle z_{B}}$ and respectively ${\displaystyle z_{C}}$ and assume that the circumcenter of triangle ABC is located at the origin of the plane. Then the complex number ${\displaystyle z_{H}=z_{A}+z_{B}+z_{C}}$

In all triangles the centroid—the intersection of the medians each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However while the orthocenter and the circumcenter are in an acute triangle’s interior they are exterior to an obtuse triangle. The orthocenter is the intersection point of the triangle’s three altitudes each of which perpendicularly connects a side to the opposite vertex . In the ca…

In all triangles the centroid—the intersection of the medians each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However while the orthocenter and the circumcenter are in an acute triangle’s interior they are exterior to an obtuse triangle. The orthocenter is the intersection point of the triangle’s three altitudes each of which perpendicularly connects a side to the opposite vertex. In the case of an acute triangle all three of these segments lie entirely in the triangle’s interior and so they intersect in the interior. But for an obtuse triangle the altitudes from the two acute angles intersect only the extensions of the opposite sides. These altitudes fall entirely outside the triangle resulting in their intersection with each other (and hence with the extended altitude from the obtuse-angled vertex) occurring in the triangle’s exterior. Likewise a triangle’s circumcenter—the interse…

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