Math 487 Lab 9

Outline of Lab

Exploring Pascal's Theorem

In this section we will do the following experiments.

1. Draw a Pascal Hexagon inscribed in a conic and drag vertices around to see what it looks like.
2. Start with 5 points and 5 lines and construct the 6^th vertex of an inscribed conic.  Then by tracing this point we will see the conic through 5 points.

Exploring Brianchon's Theorem

In this section we will do the following experiments.

3. Draw a Brianchon Hexagon circumscribed about a conic and drag the lines around to see what it looks like.
4. Start with 5 points and 5 lines and construct the 6^th vertex of an inscribed conic.  Then by tracing this point we will see the conic through 5 points.

Lab Details

Exploring Pascal's Theorem Part A

Construct a conic (say an ellipse) with the circle construction from Ogilvy that we have done before.  Be sure to make GF1 a LINE and not just a segment.  Then construct the Locus of C as G varies.  This should be an ellipse when F2 is inside the circle and a hyperbola when F2 is outside the circle.

Now make Hide/Show buttons to hide everything except the locus.

Use the point tool to place 6 points on the locus.  Name them P1, P2, …, P6 (you can do this quickly with Display > Relabel Points.

Then construct the hexagon with these 3 vertices in order using LINES and not segments.

• Let X be the intersection of P1P2 with P4P5 (two lines that you color blue).
• Let Y be the intersection of P2P3 with P5P6 (two lines that you color red).
• Let Z be the intersection of P3P4 with P6P1 (two lines that you color green). 

X, Y and Z should be collinear.  Construct the Pascal line p as line XY and make it a thick black line.

• Experiment with positions of points on the conic when the conic is an ellipse.  Note in particular ways to move the Pascal line to (approximately) to infinity.  Also notice what the figure looks like when the Pascal line is not at infinity but X is near infinity.

• Try moving pairs of points no the conic close together to give approximate tangent lines and see how the Pascal theorem could be extended to genuine tangent lines.

• Use the Hide/Show button to make the Foci, etc., visible and change the conic to be a hyperbola.  Experiment with this conic in the same way.

Exploring Pascal's Theorem Part B: Constructing a conic

In a new sketch, draw points P1, P2, P3, P4, P5 anywhere.  Then draw lines P1P2, P2P3, P3P4, P4P5, and a fifth line P5U for some new point U.  [What we really want to do is just draw any line through P5 for the fifth line, but Sketchpad requires a second point to draw the line, so we call this second point U.  U will not be a vertex of the hexagon.  It is just a control to move the line.

Now color the lines as in part A and also intersect the lines to define whichever of the points X, Y, and Z can be defined.  You will find two of them, so you can define the Pascal line p.

Now the goal is to find a point P6 on line P5X so that P1, … P6 is a Mystic Hexagon.  This means that if X, Y, Z are defined as above, they are collinear.  Denote this line by p.

Now what we have is 5 vertices and 5 sides of the hexagon.  Use the Mystic property to figure out the sixth vertex and side.

Animation of the sixth vertex of the Mystic Hexagon

When you have succeeded in constructing point P6 and thus also line P6P1, draw the conic in the following, increasingly sophisticated, ways.

• First, just trace P6 and drag U to see what the trace looks like.  It should be a conic.
• Second, animate U.  To do this, we need a path, so draw a circle c with center P5 through a new point V.  There is no special radius to us, so V can be anywhere.  Now select point U and circle c and make an Animation button.  Run the animation and observe the trace of P6.  Notice how the speed of P6 varies as U chugs around the circle.
• Third, make a locus.  This uses the "advanced optional" version of Locus with 3 givens.  First, choose P6, then U, then the circle c.  HOLD DOWN THE SHIFT KEY as you choose Construct > Locus.

Brianchon's Theorem

In this part, there will be fewer directions.

You can construct 6 tangent lines to a conic using the same construction as in the figure above, but after constructing the locus of points, construct the tangent lines by constructing  6 of the perpendicular bisectors.  These are the tangent lines.

Then construct 6 points as intersections of the 6 lines that are sides of the circumscribed hexagon.  Then connect opposite vertices with lines.  These should be concurrent no matter how you change the conic.

Challenge.  Given 5 lines and 5 points of a hexagon as above, construct the sixth side and sixth vertex so that the hexagon circumscribed a conic (you should be able to trace the sixth side to see the tangent lines of the conic.