1. Part 2
© James Taylor 2000
Dove Productions

2. Turner’s Theorem
A remarkable theorem, found in a book written by Professor Chong.
Given a triangle ABC , let there be three ellipses whose foci are
B and C,
C and A
A and B
respectively.
If we take the common chords of these ellipses (in pairs) then they are concurrent.

10. The Five Point Theorem
A conic can be drawn through any five points on the number plane.
In the following sequence, four points are kept fixed while a green point moves around the plane.

30. The Reflective Property
A ray projected from one focus of an ellipse will be reflected through the other focus.
This is often proved using an uncommon proportional theorem. The following proof uses only similar triangles.

31. y T’ P T x O S’ S
Consider the tangent at a point P on an ellipse with foci S’ and S, meeting the directrices at T’ and T as shown.

32. y T’ P T x S’ O S
A standard result gives us that  PS’T’ and  PST are 90o.

33. y T’ P T x O S’ S
Draw perpendiculars PM’ and PM as shown.
We will use the fact that PS’ = e PM’ and PS = e PM .

34. y T’ P T x O S’ S
The two coloured triangles are similar. Hence

35. y T’ P a b T x O S’ S

36. y T’ P a a T x O S’ S
So a = b, and the result is proven.

37. An Ellipse Construction
Concentric circles with radii a and b are drawn.

42. q

43. q

44. q

45. The “Off Centre” Ellipse
This construction is a variation of a paper folding envelope, where the circumference of a circle is folded over to meet a given fixed point inside the circle.

46. The centre and a point inside the circle are marked.
These points remain fixed for the construction.

47. Choose a point on the circumference.

48. Join this point to the other two.

49. Construct the perpendicular bisector as shown.
Mark the intersection with the radius.

50. Repeat with a different point on the circumference.

51. And so on.

59. The locus is an ellipse, with the two points as foci.

60. Look again at the construction.P C C’

61. G P P C C’ C’ C PC’ + PC = GP + PC = radius (which is constant)
Hence the locus is an ellipse.

62. The Kunkel Ellipse
An intriguing way of generating an ellipse, from Paul Kunkel’s excellent site.
A triangle is constrained so that two of its vertices move along two given straight lines

79. This completes the presentation
This completes the presentation. If you enjoyed the presentation or would like to make a comment, send me a note:
Paul Kunkel’s site is: