Висоти (чевіани), основи висот і ортотрикутник

Altitudes are lines perpendicular to a side of a triangle and pass through the point opposite that side.For example, in the figure below, $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are altitudes. The point that they all meet at, $H$ is called the orthocenter. They do indeed meet at one point!

The triangle formed by the feet of the altitudes (in this case $\Delta DEF$) is called the Orthic Triangle. It is also known as the Pedal Triangle.

Properties of the Orthic Triangle

1. The altitudes of the original triangle are the angle bisectors of the three angles of the orthic triangle.

2. The sides of the original triangle are the exterior angle bisectors of the angles of the orthic triangle.

3. The orthocenter, $H$, of the original triangle is the incenter of the orthic triangle.

4)The orthic triangle has the smallest perimeter of all triangles with vertices on the sides of $\Delta ABC$

Angles created by intersecting the altitudes of a triangle have some particular properties:

Suppose we have $\triangle ABC$ with feet of altitudes $E,F,$ and $G$.

Observing $\angle EBC$, we see it equals $90-\angle ACB$. Thus $\angle BHD=\angle ACB$.

Applying the same method with $\angle ABC$, we see that $\angle CHD=\angle ABC$.

Adding these two together gives $\angle BHC=\angle B+\angle C=180-\angle A$.

Using the same method for the other two angles, we conclude that

$\angle BHC=180-\angle A$ $\angle AHC=180-\angle B$ $\angle AHB=180-\angle C$

$\blacksquare$