Puzzle of 7 points and 6 straight lines – 2nd solution

This is the second solution to this puzzle: Construct a geometric shape with 7 points such that there are 6 straight lines, and each line must pass through 3 points. The first solution was already discussed last week.

And here’s the construction:

2nd solution to 7 point and 6 line puzzle

The 7 points are labelled A to G. The 6 lines are ADB, AFC, AGE, DGF, DEC and FEB. No, I didn’t label the points so I’ll have AGE and the short forms of the months December and February. It just happened that way…

The construction starts with point A, and you draw two lines down to get points B and C. The lines AB and AC must be of the same length. Then from point B, draw a perpendicular line to meet line AC. That meeting point is F.

From this construction, angles AFB, AFE, CFB and CFE are right angles (90 degrees).

Do the same thing from point C and draw a perpendicular line to meet line AB, and you’ll get point D. Similarly, angles ADC, ADE, BDC and BDE are right angles.

The point E is formed from the cross point of lines DC and FB that was just formed.

Draw a line joining D and F. Draw another line joining A and E. The cross point of lines DF and AE is point G.

The first solution focused on getting the points right, and then forming lines to fit them. This solution focused on constructing the lines, and the required points magically appear.

On hindsight, we didn’t need the right angles to be there. As long as D and F meet the lines AB and AC respectively in the same ratio, the solution is still valid. There are 2 criteria to meet:

  • Lines AB and AC must be of the same length. This allows symmetry.
  • The length ratios AD:AB and AF:AC must be equal. This is dependent on the previous criteria.

[Update] Yes, that is one heck of a correction. 3 criteria:

  • Points B, A, and C don’t form a straight line (they’re not collinear)
  • D is somewhere on the line AB (and D not equal to A nor B)
  • F is somewhere on the line AC (and F not equal to A nor C)

Then follow similar construction steps for points E and G and as Eric puts it, the rest just happens. Thank you Eric for pointing this out.

Ok, 2 solutions were presented. I hope you had fun reading and thinking about the puzzle.

  1. Eric

    Symmetry is also unnecessary. Just make sure that AB and AC meet at a non-straight angle, and make sure D is between A and B, and F is between A and C. The rest just happens.

  2. Vincent Tan

    Eric, you’re right! I completely missed it… symmetry just looks so much neater and beautiful… completely blind-sighted by it.

    Thank you for correcting me.

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