Lucky's Land

Author:
Ms. Dlouhy
Lucky buried some treasure on her land, and she left us some clues so we could find it: At the point on the map where the altitudes cross, In a hole in the ground some treasure was tossed. Stand at the place where two medians meet, And the rest of the treasure will be under your feet. Each locked in chests where they will stay Until the time someone finds a way To open each lock with a combination, Using digits of the coordinates of each location.
What do you notice about the points of concurrency of the altitudes? Why did Lucky say the treasure was where TWO altitudes meet instead of where all three meet? Do you suppose this is true for all triangles? Make your own example. We call the point of concurrency for altitudes the orthocenter. Why did Lucky say the rest of the treasure was where TWO medians meet instead of where all three meet? Do you suppose this is true for all triangles? Make your own example. We call the point of concurrency for medians the centroid. Look up the Centroid Measure Theorem. What does it claim?