Centroid, Orthocenter, Incenter, Circumcenter, Euler Line & Nine-Point Circle

This applet demonstrates the geometric constructions of a triangle's four important special points: the centroid, orthocenter, incenter, and circumcenter, as well as the Euler line and the nine-point circle. The centroid is constructed by drawing the three medians of the triangle, each connecting a vertex to the midpoint of the opposite side; their common intersection is the centroid. The orthocenter is found by constructing the three altitudes, which are perpendicular lines dropped from each vertex to the opposite side, meeting at the orthocenter. The incenter is obtained by drawing the three angle bisectors of the triangle, intersecting at the point equidistant from all three sides. The circumcenter is constructed using the perpendicular bisectors of the three sides, meeting at the center of the circumscribed circle passing through all three vertices. The applet then highlights the Euler line, a remarkable straight line that passes through the centroid, orthocenter, and circumcenter, revealing their collinearity — a classical result in triangle geometry. It also demonstrates the nine-point circle, which passes through nine significant points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments connecting each vertex to the orthocenter. The radius of the nine-point circle is exactly half the radius of the circumcircle, and its center lies on the Euler line, midway between the orthocenter and the circumcenter. Through interactive sliders and draggable vertices, users can observe how these constructions and properties are preserved under changes to the triangle's shape and size, whether scalene, isosceles, or right-angled. This dynamic visualization makes it an invaluable tool for deepening understanding of plane geometry, offering students an intuitive grasp of the elegant relationships among these classical elements. It is well suited for both classroom instruction and independent study.