In fact, this correspondence sends the whole triangle ABC to the smaller, but similar, triangle A'B'C', called the medial triangle. On one side is B, the other B' on one side C, the other C'. For each point, like A on one side of it, there is another, like A' on the other side of it but half as far away. But you might see why from the picture.įocus your attention on the centroid G. It's surprising that these three points lie on a straight line. (“Euler” is pronounced something like “Oiler” in English.) Leonhard Euler (1707–1783) was a very prolific mathematician known for his discoveries in many branches of mathematics ranging from number theory to analysis to geometry. These three “centers” of the triangle lie on one straight line, called the Euler line. For an acute triangle, the orthocenter lies inside the triangle for an obtuse triangle, it lies outside the triangle and for a right triangle, it coincides with the vertex at the right angle.įor fun, see what points and lines coincide for special triangles: isosceles triangles, right triangles, equilateral triangles, and right isosceles triangles. The altitudes of a triangle meet at a point, called the orthocenter, denoted here by H. In the case of a right triangle, two of the altitudes are actually sides of the triangle. Note that when the triangle is obtuse, two of the altitudes lie outside the triangle, so they actually connect a vertex to the opposite side extended. There are three altitudes: one is AD perpendicular to the side BC, the second is BE perpendicular to the side CA, and the third is CF perpendicular to the side AB. There's yet another interesting “center” of the triangle, the orthocenter.Īn altitude of the triangle is a line drawn through a vertex perpendicular to the side of the triangle opposite the vertex. They’re labelled A'OD', B'OE', and C'OF', and they're colored black, as are the lines connecting the midpoints of the sides, A'B', B'C', and C'A'. These are the lines perpendicular to the sides of the triangle passing through the midpoints of the sides. For acute triangles, the circumcenter O lies inside the triangle for obtuse triangles, it lies outside the triangle but for right triangles, it coincides with the midpoint of the hypotenuse.Īs Euclid proved in Propsition IV.3 of his Elements, the circumcenter can be found as the intersection of the three perpendicular bisectors of the sides of the triangle. The center of this circle is called the circumcenter, and it’s denoted O in the figure. Any three points, unless they lie on a straight line, determine a unique circle, this circumcircle. It is called the circumcircle of the triangle. You might have first noticed the circle on which the vertices of the triangle A, B, and C all lie. Only moving A, B, or C will actually change the shape of the triangle. An exception is G itself, and if you move it, the figure will slide along with it. If you drag any other point, the figure is designed to Incidentally, you can drag around other points besides the vertices A, B, and C. It can be shown that the centroid trisects the medians, that is to say, the distance from a vertex to the centroid G is twice the distance from the centroid to the opposite side of the triangle. If you make a real triangle out of cardboard, you can balance the triangle at this point. Other names for the centroid are the barycenter and the center of gravity of the triangle. This point is called the centroid of the triangle. Notice that they all meet at one point G in the triangle, also colored green. The medians of this triangle are AA', BB', CC', and they’re colored green. A line connecting a vertex of the triangle to the midpoint of the opposite side is called a median of the triangle. You can move and resize the window that appears.) The midpoint of the side BC is A', the midpoint of the side CA is B', and the midpoint of the side AB is C'. (By the way, you can place your mouse cursor over the diagram and press the return key to lift the diagram off the page. Along the sides of the triangle, you see the midpoints labelled A', B', and C'.
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