9. THE CYCLIC QUADRANGLE

Quadrangle – No Circle!  

Let A, B, C, D be four points, no three collinear. These four points determine six lines (shown in green) which occur as three pairs of opposite sides. Strictly, the points A, B, C, D form a quadrangle of which they are the vertices. However, generally the word quadrangle is used to refer to the whole figure of points and lines.

The three pairs of opposite sides determine three new points X, Y, Z, which form the vertices of the diagonal triangle of the quadrangle.

Perhaps surprisingly, there are lots of harmonic ranges associated with this diagram. In this new figure,

(U, V; Y, Z) = (D(U, V; Y, Z)) = (X, V; B, C) = (A(X, V; B, C)) = (U, V; Z, Y).

Hence by the Single Interchange Property, (U, V; Y, Z) = – 1.
You should be able to find many similar sets of harmonic points.

Is this property really surprising?
                                                                                                                                                                                                                                                                                                    
Surprising?

We have previously thought about harmonic sets (ranges) applying to points lying on circles. These results actually apply more generally to conics. We can define a conic as the locus of points which satisfy a general second degree equation: ax2 + by2 + 2fy + 2gx + 2hxy + c = 0.

In the case of the quadrangle, we may think of the four points A, B, C, D lying on a pair of opposite sides – a line pair. These form a degenerate conic, since they will have equation

(lx + my + n).(px + qy + r) = 0

an equation of the second degree. Hence it is not really surprising to find the harmonic properties occurring here.
Quadrilateral

The words quadrangle and quadrilateral are easily confused. In projective geometry, quadrangle is defined as in the text. You can think of the name as implying that the points determine the four angles (as in the case of a convex figure). On the other hand, quadrilateral speaks of four lines or edges, which determine six points. If you draw the figure, there are three extra lines dtermined by these points: the sides of a (different) diagonal triangle.

Quadrangle and quadrilateral are said to be dual concepts.

Not all authors are careful to distinguish these terms.


































Cyclic Quadrangle  Extensions 

Cyclic Quadrangle

Suppose now that the vertices A, B, C, D of the quadrangle lie on a circle. Let XYZ be the diagonal triangle as before, and let side YZ meet sides AD, BC of the quadrangle in points U, V as before.

In our previous discussion of the quadrangle, we saw that (X, V; B, C) = –1, and we could have just as easily shown that (X, U; A, D) = –1. It follows that U is the harmonic conjugate of X with respect to A and D, and V is the harmonic conjugate of X with respect to B and C.

Thinking now in terms of the circle, we deduce that U and V lie on the polar of X with respect to the circle. That is, YZ is the polar of X with respect to the circle. Similarly, XY is the polar of Z. Finally, using the reciprocal property of pole and polar, we have

Y lies on the polar of X X lies on the polar of Y,
Y lies on the polar of Z Z lies on the polar of Y,
so, YZ is the polar of Y.

A triangle for which each side is the polar of the opposite vertex is called self-polar or self-conjugate. The diagonal triangle XYZ is obviously such a triangle. Since there are an infinite number of cyclic quadrangles for a given circle, there is obviously no shortage of self-polar triangles.

                                                                                                                                                                                






























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Cyclic Quadrangle  Extensions 

Extensions

1  Given our quadrangle as usual, show that the two sides of the diagonal triangle meeting in X harmonically separate the quadrangle sides AD and BC meeting in that point.

2. Let C be a given circle and X an arbitrary point in the plane. Let Y be an arbitrary point on the polar x of X. Draw the polar y of Y to meet x in point Z. Show that XYZ is self-polar.

3. Let X be a point lying outside circle C. Use the properties of the cyclic quadrangle to give an easy way for locating the contact points of tangents from X to C.

4. Let X be a point lying inside circle C, describe a method for constructing the polar of X.
    (Hint: draw a couple of chords of the circle through X.)

5. Think about the nature of the diagonal triangle when cyclic quadrangle ABCD is in fact a rectangle.

                                                         Hints and Solutions ...
For looking up ...

 http://en.wikipedia.org/wiki/Brahmagupta's_formula

Geometry for Advanced Pupils, Maxwell, E. A. (Oxford 1963)
                                                                                                                                           
                                                                                          



































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Hints and Solutions

1  Given our quadrangle as usual, show that the two sides of the diagonal triangle meeting in X harmonically separate the quadrangle sides AD and BC meeting in that point.

• Let XY meet AB in W. Since Z is the pole of XY, (A, B; W, Z) = –1 and the result follows.

2. Let C be a given circle and X an arbitrary point in the plane. Let Y be an arbitrary point on the polar x of X. Draw the polar y of Y to meet x in point Z. Show that XYZ is self-polar.

• Let x, y, z denote polars as usual. Given Y on x, y.x = Z, so Z on x, Z on y. Hence X on y, X on z, Y on z.
   So z = XY, y = XZ, and x = YZ.

3. Let X be a point lying outside circle C. Use the properties of the cyclic quadrangle to give an easy way for locating the contact points of tangents from X to C.

• Construct a cyclic quadrangle, beginning with two lines through X which intercept C. The side x of the diagonal triangle of this quadrangle meets C at the contact points of the tangents from X.

4. Let X be a point lying inside circle C, describe a method for constructing the polar of X.
    (Hint: draw a couple of chords of the circle through X.)

• Use the hint to construct a cyclic quadrangle. The side x of the diagonal triangle of this quadrangle is the polar of X.

5. Think about the nature of the diagonal triangle when cyclic quadrangle ABCD is in fact a rectangle.

• Think of the rectangle with centre at the origin O and sides parallel to the x- and y-axes. The vertices of the diagonal triangle are O and the points at infinity determined by the (positive) axes.