I saw on a children's maths program in early 1980; I think it was Johnny Ball presenting. He drew an obtuse triangle (or I think it was obtuse). Then he put a dot anywhere in the triangle and drew three lines through the dot from each point of the triangle to the other end. He then drew a smaller triangle within the original by joining points created on the sides of the original triangle.
Before I state the problem I will try to make my explanation clearer. The dot is placed anywhere inside the obtuse triangle and a line is drawn from each of the three vertices, through the dot to the other end of the triangle. Another smaller triangle is created by drawing three lines from the points (points of intersection of the lines and the sides) on the sides of the larger triangle. Then what Mr Ball did was to find where he could project outwards from the larger triangle and the inner triangle so that the lines met, when you find the projections for all three sides of both triangles that meet, these points create a perfectly straight line.
Is there a name for this triangle projection and is there a practical use for this or is it just a fun thing to learn? I have always thought this fascinating. My maths knowledge is limited but I am always inspired by the extent of human ingenuity.
2026-03-25 14:20:05.1774448405
Can someone answer my question about geometry triangles
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1
This is Desargues's theorem. In relation to the diagram on the Wikipedia page, the point chosen in the interior of the triangle is the center of perspectivity, and the line on which all pairs of sides meet is the axis of perspectivity.
The theorem doesn't hold in complete generality in the Euclidean plane, because sides of the triangle may be parallel. It becomes true if we add "points at infinity". For every pencil of parallel lines with the same slope, we add a new point at infinity, and declare that it is on each of those lines. Further, we declare that all the points at infinity are on a single line, the "line at infinity." This construction gives the (real) projective plane, in which Desargues's theorem holds in full generality.
For example, if we have an equilateral triangle, the three pairs of sides are parallel. In the projective plane, they all intersect on the line at infinity, so Desargues's theorem holds.