I found several of the puzzles in Gamma pack to be cute, even though they aren't necessarily hard. In especially liked the "reverse" constructions, finding the triangle given the orthocenter (3.2) or given the circumcenter (3.3).

Note: toward the latter half of this pack, I was getting anxious to see what Delta pack had in store, so I bashed through constructions for 3.5-3.8 without always finding the minimal moves solutions. Among those four challenges, I still have 5 missing stars.

Triangle from orthocenter (3.2)
Obtaining the E star gave me trouble on this one. I didn't originally get one, but figured I would try harder for these notes.

The key idea is to use the E-optimal perpendicular construction from 2.6 and the vertex of the angle we're given as one of the center points. That allows us to pick up two perpendiculars for the cost of 5 E moves, leaving one last move to connect the two new vertices.

Triangle from intersection of perpendicular bisectors (3.3)

Well, I actually already gave a spoiler above when I shortened the name of this challenge. If you've got the intersection of the perpendicular bisectors, then you have the center of the orthocircle, the circle that contains all the vertices of the triangle.

Since we already have one vertex and rays where the other edges are....

Three equal distances (3.4)
When going back to write up these notes, I didn't remember how this construction worked and was concerned I'd have trouble working through a tricky challenge. Fortunately, ....

The key insight is the relationship between points B, D, and M. Just think about which of our favorite construction tricks relates them and you are done. Finding E from D and M is straightforward, but keep your eyes opened for a nice surprise!

Those V stars

Three equal distances (3.4), Forty-five degree angle (3.7), and lozenge (aka rhombus 3.8) all have V stars. In fact, 3.8 needs 4 versions to collect the V!

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It has been a long time since I worked on these. Could you remind me what the construction is for 3.5 and I'll try to give some guidance on the E solution.

3.5 is: Construct a circle through the point A that is tangent to the given line at the point B (A is not on the given line). To get the E star, you need to use 6 elementary construction steps. Is that the one you are working on?

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ReplyDeleteHa, ha. I know that this is just spam, but I appreciate the idea that India is so special that it has multiple orthocenters and one is somehow "best."

ReplyDeletehahaha

DeleteTho I still can’t prove the 5E solution to myself given the answer :-/

ReplyDeleteIt has been a long time since I worked on these. Could you remind me what the construction is for 3.5 and I'll try to give some guidance on the E solution.

Delete3.5 is: Construct a circle through the point A that is tangent to the given line at the point B (A is not on the given line). To get the E star, you need to use 6 elementary construction steps. Is that the one you are working on?

Delete