Recalling the Four Triangles, the circumcenters of these triangles are concyclic, lying on the Circumcentric Circle, whereas the orthocenters are collinear, lying on the Orthocentric Line. Construct the perpendicular bisector of each circumcenter/orthocenter pair for each triangle. These perpendicular bisectors are concurrent at the Hervey Point.
Tuesday, March 23, 2010
Monday, March 22, 2010
The Complete Quadrilateral: Even More Points on The Orthocentric Line
Are you familiar with the Orthopole? I first ran across this at a presentation by my friend Ray Klein, a T3 National Instructor from the Chicago area. If you are not familiar with it, the following applet will illustrate the construction.
In the Complete Quadrilateral, construct the Orthopoles of the Four Triangles. The Orthopoles lie on the Orthocentric Line.
In the Complete Quadrilateral, construct the Orthopoles of the Four Triangles. The Orthopoles lie on the Orthocentric Line.
Sunday, March 21, 2010
The Complete Quadrilateral: More Points on the Orthocentric Line
Using pairs of opposite sides, drop perpendiculars from the midpoint of one to the other. These pairs of perpendiculars intersect in six points total, which lie on the Orthocentric Line.
Saturday, March 20, 2010
The Complete Quadrilateral: The Pedal Line, The Orthocentric Line, and The Focal Point
This makes a pretty picture. Drag the slider slowly to the left to reveal each line or point one at a time. If you do anything with conics and/or their constructions, this is an unexpected conic, which are the best kind!
Draw the Pedal Line. Recall the Pedal Line is the line through the feet of the perpendiculars dropped from the Focal Point to each side. Recall the Focal Point is the intersection point of the four circumcircles of the Four Triangles.
Draw lines through the midpoints of pairs of opposite sides.
These seven lines, along with the four lines of the quadrilateral, are tangent to the parabola whose focus is the Focal Point and whose directix is the Orthocentric Line.
Draw the Pedal Line. Recall the Pedal Line is the line through the feet of the perpendiculars dropped from the Focal Point to each side. Recall the Focal Point is the intersection point of the four circumcircles of the Four Triangles.
Draw lines through the midpoints of pairs of opposite sides.
These seven lines, along with the four lines of the quadrilateral, are tangent to the parabola whose focus is the Focal Point and whose directix is the Orthocentric Line.
What Does This Say About Grades and Grading?
Granted, it is a game design class, but could the same thing be applied to other education settings? Followed a link in the above article to this, conaining the following quote
"The elements of the class are couched in terms they understand, terms that are associated with fun rather than education," [Sheldon] told iTnews. "There will always be a portion of the class who will not be motivated to learn no matter what an instructor may try. Those that are not as involved, one or two out of a class of forty, are pretty much drifting through life anyway thanks to factors the classroom can't really address."
I buy that!
Friday, March 19, 2010
The Complete Quadrilateral: Circles on Diagonals
Construct the diagonals of the quadrilateral, and then construct circles with the diagonals as diameters. The centers of the circles are the midpoints of the diagonals, and we know they lie on the Midline. The three circles intersect in two distinct points on the Orthocentric Line. Therefore, the Midline and the Orthocentric Line are perpendicular. In circle-talk, these three circles are coaxal, and the Orthocentric Line is the radical axis of the three circles.
The Complete Quadrilateral: The Orthocentric Line
Construct the Orthocenters of the four triangles.
Before dragging the slider, make a prediction. What will be true about the four orthocenters?
Before dragging the slider, make a prediction. What will be true about the four orthocenters?
Thursday, March 18, 2010
More Wallace-Simpson Line
I guess I should elaborate a bit more on the Wallace-Simpson Line, considering it is used to make a picture that produces the same reaction every time I show it to my kids: THAT is frickin' COOL!
The envelope of every Wallace-Simpson line for every point on the circumcircle is the Steiner deltoid whose area is half the area of the circumcircle. The Steiner deltoid is also tangent to the Nine-Point Circle, the unique circle that passes through the midpoints of the sides of the triangle.
Click on the small animation button in the lower left corner. Click on the reset icon in the upper right corner erase the traces.
The envelope of every Wallace-Simpson line for every point on the circumcircle is the Steiner deltoid whose area is half the area of the circumcircle. The Steiner deltoid is also tangent to the Nine-Point Circle, the unique circle that passes through the midpoints of the sides of the triangle.
Click on the small animation button in the lower left corner. Click on the reset icon in the upper right corner erase the traces.
An Integrated Curriculum?
Might this fall under the category of WCYDWT?
I am the chess sponsor at my school. I play, and I can beat some of the kids in my club some of the times, but generally, I just provide a place to play after school, and I drive the school van to the matches. We actually did OK this past year, finishing 2nd in our division, 2nd in the league tournament, and 1st in the league playoffs.
Anyway, since all the of kids who play are big UFC fans, I had joked with them how if we happened to lose the match, we could win the fight afterwords. And then I am up the other morning watching ESPN and THIS comes on
I am the chess sponsor at my school. I play, and I can beat some of the kids in my club some of the times, but generally, I just provide a place to play after school, and I drive the school van to the matches. We actually did OK this past year, finishing 2nd in our division, 2nd in the league tournament, and 1st in the league playoffs.
Anyway, since all the of kids who play are big UFC fans, I had joked with them how if we happened to lose the match, we could win the fight afterwords. And then I am up the other morning watching ESPN and THIS comes on
Wednesday, March 17, 2010
The Complete Quadrilateral: The Pedal Line
This theorem reminds me of the Wallace-Simpson line in a triangle. The Wallace-Simpson line is constructed by dropping perpendiculars to the sides (extended) of a triangle from a point on the circumcircle. The feet of these perpendiculars lie on a line. This line is the Wallace-Simpson line.
In the complete quadrilateral, from the Focus Point, we drop perpendiculars to the sides of the quadrilateral. The feet of these perpendiculars are also collinear. This is known as the Pedal Line. The proof of this follows directly from the Wallace-Simpson line.
In the complete quadrilateral, from the Focus Point, we drop perpendiculars to the sides of the quadrilateral. The feet of these perpendiculars are also collinear. This is known as the Pedal Line. The proof of this follows directly from the Wallace-Simpson line.
The Complete Quadrilateral: Connecting the Midpoints of Opposite Sides
When you look at the six pairs of opposite sides, you may notice that the pairs themselves could be grouped into three sets of four that "look alike."
For example, the opposite sides formed by segment A12A14 and segment A23A34 seem to naturally go together with the opposite sides formed by segment A12A23 and segment A14A34.
Likewise, the opposite sides formed by segment A23A24 and segment A13A14 seem to naturally go together with the opposite sides formed by segment A13A23 and segment A14A24.
When you construct segments connecting the midpoints of these three sets of opposite sides, the segments intersect in collinear points.
For example, the opposite sides formed by segment A12A14 and segment A23A34 seem to naturally go together with the opposite sides formed by segment A12A23 and segment A14A34.
Likewise, the opposite sides formed by segment A23A24 and segment A13A14 seem to naturally go together with the opposite sides formed by segment A13A23 and segment A14A24.
When you construct segments connecting the midpoints of these three sets of opposite sides, the segments intersect in collinear points.
Tuesday, March 16, 2010
The Complete Quadrilateral: The Axis of Mean Distances
Construct the midline of the Complete Quadrilateral. Drag the slider slowly to the left to reveal the significance of the Axis of Mean Distances.
How would this change if two sides of the quadrilateral are parallel?
How would this change if two sides of the quadrilateral are parallel?
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