from The American Heritage® Dictionary of the English Language, 4th Edition
- n. The point of intersection of the three altitudes of a triangle.
from Wiktionary, Creative Commons Attribution/Share-Alike License
- n. : the intersection of the three lines that can be drawn flowing from the three corners of a triangle to a point along the opposite side where each line intersects that side at a 90 degree angle; in an acute triangle, it is inside the triangle; in an obtuse triangle, it is outside the triangle.
from the GNU version of the Collaborative International Dictionary of English
- n. That point in which the three perpendiculars let fall from the angles of a triangle upon the opposite sides, or the sides produced, mutually intersect.
from The Century Dictionary and Cyclopedia
- n. The cointersection point of the straight lines through the three vertexes of a triangle perpendicular to the opposite sides.
Sorry, no etymologies found.
The first is called the orthocenter, and is the intersection of the lines from each vertex that meet the opposing sides perpendicularly, which are called the altitudes.
In 1767 Leonhard Euler proved that for all triangles the orthocenter, the circumcenter, the centroid and the center of the midcircle are always on the same line.
Yet not until 1820, when a paper ∗ published by Brianchon and Poncelet appeared, were the remaining three points (the midpoints of the segments from the orthocenter to the vertices) found to be on this circle.
Geometric Wonders 185 C B 'A' A B C 'F Figure 5. 23b With H as the orthocenter (the point of intersection of the altitudes), M is the midpoint of CH (see Figure 5.23d).
These nine points, for any given triangle, are • The midpoints of the sides • The feet of the altitudes • The midpoints of the segments from the orthocenter to the vertices Have your students do the necessary construction to locate each of these nine points.
Theorem In any triangle, the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the orthocenter to the vertices lie on a circle.
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