# Sketchpad Investigations of the Orthocenter

#### A few definitions to begin with:

 orthocenter - the intersection of the three altitudes of a triangle. centroid - the intersection of the three medians of a triangle. If a triangle could have constant area mass density, the centroid would be the center of mass, or balancing point. circumcenter - the intersection of the perpendicular bisectors of the three sides of a triangle. The circumcenter is the center of the circumscribed circle. incenter - the intersection of the bisectors of the three interior angles of a triangle. The incenter is the center of the inscribed circle.

 1 Open a new sketch, and construct a triangle. Label the vertices A, B, and C. 2 Construct all three altitudes. To do this, select one vertex and the opposite side. From the Construct menu, choose Perpendicular Line. Repeat these steps with the other two vertices. 3 The three altitudes should be concurrent. Why? Sketchpad cannot define the intersection point of three lines, so select any two of the altitudes. From the Construct menu, choose Point At Intersection. Label the orthocenter D. 4 Drag the triangle vertices, and observe the position of the orthocenter. Under what conditions is it on the interior of the triangle? When is it on the exterior? When is it on a side of the triangle? When is it on a vertex? 5 Construct the line segments DA, DB, and DC. Hide the altitude lines. There are now three triangles on the sketch, ΔABC, ΔDAB, ΔDBC, and ΔDCA. You have already constructed the orthocenter of ΔABC. Where are the orthocenters of the other triangles?

The points A, B, C, and D are called an orthocentric system. In this activity, you will explore some of the special properties of the system. It was drawn on the plane, but it may be helpful to imagine the four triangles as the four faces of a tetrahedron.

 6 Before going any further, select all four points and all six sides. From the Edit menu, choose Action Button, Hide/Show. The two buttons that appear on the screen will help you organize the drawing if it gets cluttered, which it will. It also would help to give the segments some color. Select the six segments. From the Display menu, choose Color, and select a color. 7 Construct the midpoints of all six segments. To do this, select the segments, and from the Construct menu, choose Point At Midpoint. 8 Use the midpoints to construct the perpendicular bisectors of the sides; then use the perpendicular bisectors to construct the circumcenters of all four triangles. Use the text tool to label the four circumcenters. The circumcenter of ΔABC should be labeled D', the circumcenter of ΔDAB should be labeled C', and so on. The reason for this will become clear later. 9 Hide the bisectors, and use the circumcenters to construct the circumcircle of each of the four triangles. What do you observe about the circles? Can you explain their relation? Those circles might be useful later. Select all four of them, and create Hide/Show buttons. Hide the circles for now. 10 From the Display menu, choose Color. Select some color other than the last one you used. Now construct line segments connecting each pair of the points in A', B', C', and D'. What do you observe? 11 This new system appears to be congruent to the first one. Can you support this with measurements? Can you prove it? 12 In fact, the systems are congruent. They also are 180° rotations of each other. A 180° rotation is the equivalent of a point reflection. Construct lines between corresponding points (A and A', B and B', etc). They should intersect at the reflection point. 13 The reflection point has another special property. Choose the circle tool from the tool bar. Click on the reflection point and drag to one of the segment midpoints. This is called the nine-point circle. It includes six of the points identified on this sketch. See if you can find the other three (actually, six). How does the nine-point circle compare with the circumcircles that were constructed earlier? 14 There is still more. If ΔABC is not acute, drag the vertices so that it is. On the Display menu, set the line style to draw a thick, black line. From the tool bar, choose the segment tool. Go around the circle in order, connecting each pair of adjacent midpoints with a line segment. Next, connect point D with each of its adjacent midpoints. Do the same with D'. The black segments have drawn a projection of a rectangular solid. Drag the vertices again. Why does this not work when ΔABC is obtuse?

#### Extensions:

After constructing the orthocentric system, you constructed the circumcenter of each triangle. Try it again with the centroids. Try it with the incenters.

Go back to the orthocentric system again. When point A is dragged, points B and C do not move, but point D does. What shape is traced by point D when point A moves along a line? What happens when Point A moves along a circle?

#### Isogonal Transformation

Construct a triangle, and bisect its interior angles. Place another point in the picture, not necessarily in the triangle. Construct lines from this point to each vertex of the triangle. Reflect each of those lines across its corresponding angle bisector. The three reflections will be concurrent. Their point of intersection is the isogonal transformation of the first point. Where is the isogonal transformation of the orthocenter of the triangle? What special property does it have?