Immanent Triangles

Immanent Triangles is one of two related phenomena which have hitherto gone unobserved in the literature in geometry (the other is Immanent Squares). This phenomenon allows the construction of equilateral triangles from irregular ones and illustrates the presence of the perfect in the imperfect. Related to Immanent Triangles is a wealth of other phenomena some of which are included.

Fig.1

 

  1. If we draw a scalene triangle ABC and

  2. construct an equilateral triangle above each one of its three sides

  3. the centre points of these three triangles L1M1N1 will always

  4. determine an equilateral triangle (we will call it immanent triangle).

 

Fig.2 

 

  1. follows the same process but the three equilateral triangles are folded in. The result is again an equilateral triangle, but smaller and differently positioned.

Fig.3 

 

shows the larger immanent triangle

 

  1. scalene triangle

  2. rectangular triangle

  3. isosceles triangle

  4. equilateral triangle

  5. triangle in one line

  6. triangle with point C at infinity

  7. triangle in one line and point C and B coinciding

Fig.4 
 

shows the smaller 'immanent' triangle in the same sequence (shrinks to a point in d).

 

  1. scalene triangle

  2. rectangular triangle

  3. isosceles triangle

  4. equilateral triangle

  5. triangle in one line

  6. triangle with point C at infinity

  7. triangle in one line and point C and B coinciding

Fig.5  Reoccurring Six Star on Triangle

 

  1. Shows another beautiful phenomena arising from the same construction. ABC is a scalene triangle. Three equilateral triangles are folded out on its sides. The results are three new points D1E1 and F1. If we connect opposite points AF1, BD1 and CE1, the three resulting lines will always meet in one point M and form a perfect 6-ray star.= the larger immanent star.

  2. Shows the same phenomena arising out of its infolded counterparts D2E2F2= smaller immanent star.

Fig.6 Elaborates on correspondences between folding in and out.

 

  1. ABC is a scalene triangle and L1M1N1 and L2M2N2 are two immanent triangles. D1E1F1 are the corner points of the three equilateral triangles folded out.

  2. D1E1 and F1 are connected into a triangle.

  3. From the side D1F1 an equilateral triangle is folded in →G1. The centre point of this equilateral triangle coincides with the point N2 of the smaller immanent triangle.

  4. As equilateral triangles are folded in from all three sides of the triangle D1E1F1 their centre points coincide with the corner points of the smaller immanent triangle L2M2N2.

 

Fig.7 shows the opposite process

 

  1. Here the result of the first folding in, the triangle D2E2F2 is folded out. The three new centre points coincide with the large immanent triangle L1M1N1.

Fig.8

 

  1. ABC is the initial scalene triangle. D1E1F1 are the components of the three equilateral triangles on ABC. Now three equilateral triangles are folded in on D1E1F1. Their corner points G1H1J1 are connected and result in a 'parallel triangle' circumscribing ABC.

 

  1. shows its 'symetric' equivalent. The first infolded triangle D2E2F2 is folded out. The corner points form the same parallel triangle circumscribing ABC as in Fig.5a. G1H1J1 and G2H2J2 are identical.