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Triangle Applet

This interactive Java applet is demonstrating some fundamental properties of plane triangles. Check radio buttons and click to set the three vertices A, B, and C. Uncheck all the other checkboxes. 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 nine-point 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 seven-point-circle, 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.

 Web Links Triangle (Wikipedia) Encyclopedia of Triangle Centers (Wikipedia)

Letzte Änderung 09.08.2018