In mathematics people say that two triangles ABC and A'B'C' are (in) perspective, if the lines AA', BB' en CC' come together in one point P. This point P is called the center of perspective or perspector.
In October 1997 I constructed, while playing around with triangles, an amazingly nice theorem. "Is this theorem known?" I thought at that moment. Maybe not in my wording, but the figure we end in was already known.
Nederlandse versie van deze pagina.
|
In mathematics people say that two triangles ABC and A'B'C' are (in) perspective, if the lines AA', BB' en CC' come together in one point P. This point P is called the center of perspective or perspector. |
|
About perspective triangles there are loads of known theorems but nobody seems to know the theorem that I discovered in October 1997. For this theorem we first construct the cross-triangle.
|
Out of two triangles ABC and A'B'C' the cross-triangle A"B"C" can be constructed in the following way: A" is the point of intersection of BC' and B'C B" is the point of intersection of AC' and A'C C" is the point of intersection of AB' and A'B |
|
|
The first part of the theorem about the cross-triangle A"B"C" that I've proved is the following: If ABC en A'B'C' were perspective, so are ABC en A"B"C". The center of perspective we call Q. |
|
|
The second part of the theorem is a logical follow-up, that the triangles A'B'C' and A"B"C" are also in perspective. The center of perspective we now call R. |
|
The end of the theorem is that the three centers of perspective that we have found lie on one straight line. 
I will not present a full proof here now. But the proof isn't very difficult. If you consider the triangles ABC and A'B'C' as projections of a pyramid with top P (or, if necessary as a double pyramid), then the theorem follows from some simple manipulations of planes.
This theorem and its proof have been published in:
Floor van Lamoen, Bicentric triangles, Nieuw Archief voor Wiskunde, 17-3 363-372 (1999).
Back to Floors wiskunde pagina.
Back to Home.
