The orthocenter of $\Delta ABC$ coincides with the circumcenter of $\Delta A'B'C'$ whose sides are parallel to those of $\Delta ABC$ and pass through the vertices of the latter. The foot of an altitude also has interesting properties. For example, due to the mirror property the orthic triangle solves Fagnano's Problem. I have collected several proofs of the concurrency of the altitudes, but of course the altitudes have plenty of other properties not mentioned below. Let's observe that, if $H$ is the orthocenter of $\Delta ABC$, then $A$ is the orthocenter of $\Delta BCH,$ while $B$ and $C$ are the orthocenters of triangles $ACH$ and $ABH,$ respectively. It is listed below, but appears on a separate page along with historical remarks. The earliest known proof was given by William Chapple (1718-1781). The timing of the first proof is still an open question it is believed, though, that even the great Gauss saw it necessary to prove the fact. In our triangle here in the above diagram. It is also worth noting that the position of the orthocenter changes depending on the type of triangle for a right triangle, the orthocenter is at the vertex containing the right angle for an obtuse triangle, the orthocenter is outside the triangle, opposite the longest side for an acute triangle, the orthocenter is within the triangle.This is a matter of real wonderment that the fact of the concurrency of altitudes is not mentioned in either Euclid's Elements or subsequent writings of the Greek scholars. An altitude is basically a perpendicular line segment that is drawn from a vertex of a triangle to the opposite side. Along with the use of trigonometric relationships, the altitudes of a triangle can be used to determine many characteristics of triangles. The intersection between the extended base and the altitude is called the foot of the altitude. This line containing the opposite side is called the extended base of the altitude. forming a right angle with) a line containing the base (the opposite side of the triangle). Each of the altitudes of a triangle forms a right triangle, and the altitudes of a triangle all intersect at a point referred to as the orthocenter. In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e. The base of a triangle is determined relative to a vertex of the triangle the base is the side of the triangle opposite the chosen vertex. Since all triangles have 3 vertices, every triangle has 3 altitudes, as shown in the figure below:
![efine altitude geometry efine altitude geometry](https://www.mathopenref.com/images/general/parallelbase.gif)
An altitude of the isosceles triangle is shown in the figure below: Thus, we can conclude that a median is drawn from a vertex of the triangle to the midpoint of the opposite side and divides the opposite side into two equal parts or halves, whereas an altitude is drawn from a vertex of the triangle to the opposite side being perpendicular to it. a Z coordinate, denoting altitude an M coordinate (rarely used).
![efine altitude geometry efine altitude geometry](https://study.com/cimages/multimages/16/altitude1.png)
In other words, an altitude in a triangle is defined as the perpendicular distance from a base of a triangle to the vertex opposite the base. Spatial attributes are geometry valued, and simple features are based on 2D geometry. In a triangle however, the altitude must pass through one of its vertices, and the line segment connecting the vertex and the base must be perpendicular to the base.
![efine altitude geometry efine altitude geometry](https://mathmonks.com/wp-content/uploads/2021/05/Properties-of-Altitude-of-a-Triangle.jpg)
The height of an object like an aeroplane or kite, or of a place is equal to its height above Toggle. In other geometric figures, such as those shown above (except for the cone), the altitude can be formed at multiple points in the figure. Definition of Altitude Altitude is another word for height. Altitude in trianglesĪltitude in triangles is defined slightly differently than altitude in other geometric figures. Note that the altitude can be depicted at multiple points within the figures, not just the ones specifically shown.
![efine altitude geometry efine altitude geometry](https://image.slidesharecdn.com/8-150213135642-conversion-gate01/95/87-angles-of-elevation-and-depression-5-638.jpg)
The dotted red lines in the figures above represent their altitudes.