Triangle Applet
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Then check radio button and click to
set a pedal point P of triangle ABC. From point P perpendiculars are droped to the three sides of the triangle. If the pedal point is on the circumcircle of the triangle, A'B'C' collapses to a line. This is then called the pedal line, or sometimes the Simson line. 

An altitude of a triangle is a
straight line through a vertex and perpendicular to
the opposite side The three altitudes of a triangle intersect at the orthocenter H. 

The pedal triangle of the orthocenter of the original triangle is called the orthic triangle or altitude triangle.  
The median of a triangle is a line
segment joining a vertex to the midpoint of the
opposing side. Every triangle has exactly three
medians, one from each vertex, and they all intersect
each other at the triangle's centroid. The medial triangle or midpoint triangle is the triangle with vertices at the midpoints of the triangle's sides. The Spieker center S of triangle ABC is the incenter of the medial triangle. The Spieker center lies on the Nagel line, and is therefore collinear with the incenter, triangle centroid, and Nagel point. The touch points of the excircles of ABC connected to the opposite vertices of ABC concur in the Nagel point N. 

The perpendicular side bisectors are passing through the midpoint of the side and being perpendicular to it. They meet in a single point, the triangle's circumcenter  
The circumcircle of a triangle is a circle which passes through all the vertices of the triangle. The center of this circle is called the circumcenter.  
The ninepoint circle passes
through nine significant concyclic points defined
from the triangle: ¤ the midpoint of each side of the triangle, ¤ the foot of each altitude, ¤ the midpoint of the line segment from each vertex of the triangle to the orthocenter (where the three altitudes meet); these line segments lie on their respective altitudes. 

The incircle (inscribed circle)
of a triangle is the largest circle contained in the
triangle touching the three sides. The three lines connecting a vertex and the opposite touchpoint of the incircle intersect in the Gergonne point. 

An excircle of the triangle is a
circle lying outside the triangle, tangent to one of
its sides and tangent to the extensions of the other
two. Every triangle has three distinct excircles,
each tangent to one of the triangle's sides. The Mandart circle is the circumcircle of the extouch triangle. 

The Euler line of a (non equilateral) triangle passes through several important points including the orthocenter, the circumcenter, the centroid, the Exeter point, and the orthocenter of the anticomplementary triangle.  
The Steiner ellipse of a
triangle is the unique circumellipse (touching the
triangle at its vertices) whose center is the
triangle's centroid. It is the ellipse of least area
that passes through A, B, and C. The Steiner inellipse is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It has the maximum area of any inellipse. The Steiner inellipse has semiaxes lengths which are half of the circumellipse. 

The anticomplementary triangle
is the triangle which has a given
triangle as its medial triangle. The de Longchamps point (the reflection of the orthocenter about the circumcenter) is the orthocenter of the anticomplementary triangle. 

The three blue tangent circles are
centered at A, B, and C, and are pairwise tangent
to one another. There exist exactly two
nonintersecting circles that are tangent to all
three circles: the inner and outer Soddy
circles. The Soddy center P is the isoperimetric point of ABC, having the property that the triangles PBC, PCA and PAB have isoperimeters. 

There are an infinitude of lines that
bisect the area of a triangle. Three of them are
the medians of the triangle, and these are
concurrent at the triangle's centroid. Three other
area bisectors are parallel to the triangle's
sides; each of these intersects the other two
sides so as to divide them into segments with the
proportions 1 : sqrt(2)+1. These
six lines are concurrent three at a time 

The line that divides the angle into two
equal parts is known as the angle bisector. Three angle bisectors of a triangle meet at a point called the incenter of the triangle. 

On the sides of the triangle, construct
outwardly or inwardly drawn equilateral triangles.
The lines connecting the centroids of these
triangles to the opposite vertex of the triangle
intersect at the Napoleon points.
The centroids form an equilateral triangle. 

The angles subtended by the sides of the
triangle at the Fermat point are
all equal to 120°. The total distance from the three vertices of the triangle to this point is the minimum possible. 

The triangle A'B'C'
is formed by the tangents at A, B,
and C to the circumcircle of
triangle ABC. The lines passing
the vertices of triangle
A'B'C' and the
intersection points A'', B'', and C'' of the medians
of ABC with the circumcircle, will meet at the Exeter
point, lying on the Euler line. 

There is exactly one point P such that the line segments AP, BP, and CP form the same angle, Brocard angle ω, with the respective sides c, a, and b.  
The point of intersection of BP (P
Brocard point 1) and CQ (Q Brocard point 2) is B1,
similarly B2, and B3. B1B2B3 is the Brocard triangle. Its circumcircle is the Brocard circle or sevenpointcircle, containing B1, B2, B3, P, Q, K (symmedian point), and the center of the circumcircle of ABC. The symmedian point K is the point of concurrence of the symmedians, constructed by reflecting the medians of ABC about the angle bisectors. 

Triangle (Wikipedia) Encyclopedia of Triangle Centers (Wikipedia)

Letzte Änderung 21.10.2023