1. Draw an acute (scalene) triangle ABC on patty paper.

• Construct a median in triangle ABC from vertex A, creating segment AD
• Construct a median in triangle DAB from vertex D, creating segment DE
• Construct a median in triangle DAC from vertex D, creating segment DF
• List as many conjectures as you can
• Compare your results with those of your partners and try to verify or disprove your conjectures by making measurements.

2. Draw a new acute triangle ABC, construct three altitudes (to D, E, and F) and label their point of intersection G.

• Reflect point G in each of the three sides of triangle ABC, producing points H, I, J.
• Draw triangle DEF and triangle HIJ
• Write as many conjectures as you can about the relationships among points, segments, angles, and triangles.
• Compare your results with those of your partners and try to verify or disprove your conjectures.

3. Draw a new triangle ABC, construct three altitudes (to D, E, and F) and label their point of intersection G (this is the orthocenter). (Make sure some people make non-acute triangles for this experiment.)

• Draw segments from the orthocenter to each of the vertices.
• Find the orthocenter of each of the new triangles formed.
• Compare with your partners and make observations or conjectures.

1. Draw an acute (scalene) triangle ABC.

• Construct a median in triangle ABC from vertex A, creating segment AD
• Construct a median in triangle DAB from vertex D, creating segment DE
• Construct a median in triangle DAC from vertex D, creating segment DF
• List as many conjectures as you can
• Drag vertices of your triangle and verify or disprove your conjectures by making measurements.
• Write as many statements as you can that will always be true of acute scalene triangles
• Write as statements as you can that will always be true provided that you start with a particular kind of triangle (e.g. right or isosceles).

2. Open a new sketch. On an acute triangle ABC, construct three altitudes (to D, E, and F) and label their point of intersection G.

• Reflect point G in each of the three sides of triangle ABC, producing points H, I, J.
• Construct triangle DEF and triangle HIJ
• Write as many conjectures as you can about the relationships among points, segments, angles, and triangles.
• Drag vertices of triangle ABC and use measurements to verify or disprove your conjectures.

3. Open a new sketch and choose Preferences in the Display menu. Choose more (options) and set your script tools folder to Triangle Special Points under scripts in the samples folder.

• Chose orthocenter from the script tool.
• Click on three different locations to construct triangle ABC and its orthocenter.
• Drag a vertex of the triangle and make observations about when the orthocenter is inside, outside, or on the triangle.
• With the orthocenter on the inside of the triangle, construct segments from the orthocenter to each of the vertices.
• Play the orthocenter script on each of the new triangles formed.
• Experiment with dragging different points. Make observations and conjectures about orthocenters.