Construct the Orthocenters of the four triangles.
Before dragging the slider, make a prediction. What will be true about the four orthocenters?
Friday, March 19, 2010
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?
Monday, March 15, 2010
The Complete Quadrilateral: The Mid-Diagonal Line
The next stop on our Complete Quadrilateral tour is the Mid-Diagonal line, or the Midline. This line goes by many names. Ripert called it the Axis of Mean Distances (which we will see why in the next stop on the tour). It is also known as the Newton-Guass line, or the Newtonian. Whatever it is called, it is one of the earliest known lines associated with the Complete Quadrilateral.
We begin by drawing the diagonals of the quadrilateral, and constructing their midpoints. Two of the diagonals will be familiar (for example, the A12 to A34 diagonal). One diagonal - from A13 to A24 - will seem a bit odd. Nevertheless, their midpoints are collinear, and this line is the Midline.
We begin by drawing the diagonals of the quadrilateral, and constructing their midpoints. Two of the diagonals will be familiar (for example, the A12 to A34 diagonal). One diagonal - from A13 to A24 - will seem a bit odd. Nevertheless, their midpoints are collinear, and this line is the Midline.
Saturday, March 13, 2010
Function Families
During the fourth quarter, I do a "pre-Algebra 2" unit on function families. One of the results I would like my students to see is that any function they will encounter in Algebra 2 can be written in a form similar to the "slope" form I use with my Geometry students.
When we study similar triangles, we go back and visit the equation of a line that they learned in Algebra 1 and look at it through slope eyes.
Consider a line through the points (x1,y1) and (x2,y2). Place an arbitrary point (x,y) anywhere on the line. I know that the slope triangles shown in the graph are similar, hence, the ratios of the corresponding sides are equal. In other words, the slopes as measured by these two point are equal.
Being this is the case, we can write an equation stating the two slopes are equal
The remarkable thing is we can write the equation of an absolute value graph using a similar form. If the vertex of the graph is at (x1,y1) and (x2,y2) is another known point on the graph, the equation of the absolute value graph can be written in the form
Likewise, a quadratic with a vertex at (x1,y1) and (x2,y2) another known point on the graph can be written as
Similarly, the equation of cubic (the transformed parent function y = x3 ) can be written as
Only slight modifications are needed to apply this cubic equation form to any cubic function, or to an exponential function, or even to a periodic function.
In essence, the point-slope form of a line from Algebra 1 can be modified to fit three other function families: Absolute Value, Quadratic, and Cubic.
Try using these equation forms in the applet below. Use point A for the vertex and point B for the other point on the graph. Calculate the rise and the run as you would for a linear equation. For calculating convention, your rise should calculated using yB - yA, provided point A is your vertex. Enter your equation in the input line. Click on the RESET ICON in the upper left corner of the applet to create a new pair of points.
Friday, March 12, 2010
The Complete Quadrilateral: Perpendicular Bisectors of Opposite Sides
Recall the opposite sides of a complete quadrilateral, and construct the perpendicular bisectors of each pair. Provided the bisectors are not the same line, the perpendicular bisectors of opposite sides will intersect at either a real point or at an ideal point (when the two perpendicular bisectors happen to be parallel). Construct the six pairs of perpendicular bisectors and their intersection points.
Drag the GREEN slider slowly to the left, revealing the bisectors and intersection points one at a time.
I will leave the theorem for you to state.
Drag the GREEN slider slowly to the left, revealing the bisectors and intersection points one at a time.
I will leave the theorem for you to state.
The Complete Quadrilateral: The Circumcentric Circle
Construct the circumcenters of each of the four triangles. The four circumcenters lie on a circle. However, there is more to this circle than that. Slowly drag the PURPLE slider to the left to reveal the centers one by one. As you reveal the circumcenters, can you guess what will happen?
Once again, what will happen when two of the sides are parallel? What about two pairs of sides being parallel?
Once again, what will happen when two of the sides are parallel? What about two pairs of sides being parallel?
Thursday, March 11, 2010
The Complete Quadrilateral: The Focal Point
John Wentworth Clawson calls this point the Focal Point. This point is also known as the Wallace Point and the Miquel Point. For each of the four triangles in the quadrilateral, construct the circumcircle. In the sketch below, the conclusion is rather evident. Slowly drag the PURPLE slider to the left to reveal each circumcircle, one at a time, until all are shown.
Play around with this a little bit. What happens when two sides of the quadrilateral are parallel? What happens when two pairs of sides are parallel?
Play around with this a little bit. What happens when two sides of the quadrilateral are parallel? What happens when two pairs of sides are parallel?
The Complete Quadrilateral: Four Triangles
There are four triangles that will also play a part in our investigation. I am not sure about you, but when I see a whole bunch of triangles related to each other in a systematic way, I begin to think about how their centers might be related.
Drag the RED slider slowly to the left to reveal the four triangles one at a time.
Drag the RED slider slowly to the left to reveal the four triangles one at a time.
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