Given the orthocenter and circumcenter of a triangle along with a line on which one of the sides of the triangle lie, how many moves are required to draw that triangle using a straight-edge and collapsible compass ?
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I have a 15 move solution.
Let H and O be the orthocentre and circumcentre respectively, and the line as l .
First, we will draw perpendiculars from O and H onto l . The method for drawing perpendiculars from a point P onto a line p is as follows:
From an arbitrary point Q on p , draw a circle passing through P . (1 move)
From another arbitrary point R (different from Q) on p , draw a circle passing through P . (1 move)
These two circles intersect at P and another point P ′ . Draw P P ′ . This is perpendicular to p . (1 move)
Therefore, it takes 3 moves to construct a perpendicular from an exterior point onto a line, so 6 moves will be used to construct the two perpendiculars.
Now, we will locate the centroid (call it G ). By the property of the Euler line, we have H G : G O = 2 : 1 . Thus, we will trisect O H partially.
Draw O H (and extend it). (1 move)
Draw a circle centred at O passing through H . Let this intersect O H again at X . (1 move)
Draw a circle centred at H passing through X . Let this intersect O H again at Y . (1 move)
Draw a circle centred at Y passing through O . Let this intersect the circle centred at O passing through H at Z . (1 move)
Draw a circle centred at Z passing through O . This intersects O H again at G . (1 move)
Thus, we have the centroid G located. Let the foot of the perpendicular from O l be called M and the foot of the perpendicular from H to l be called N . Thus, G M is the median of the triangle. We will now locate the apex vertex.
Thus, we have located the apex vertex. We will finally locate the other two vertices and draw the sides.
Draw a circle centred at O passing through A . Let it intersect l at B and C . (1 move)
Draw A B and A C . (2 moves)
Thus, we have drawn the triangle, and it took 6+5+4=15 moves.
Jon Haussman has also got 15 moves:
Construct M and N as usual; this takes 6 moves. This construction also gives us line H N .
∙ Draw the circle centred at H with radius H O . (1 move)
∙ Draw the circle centred at O with radius H O . (1 move)
∙ Draw the line H O . (1 move)
∙ Let the two circles just drawn intersect at X and Y . Draw X Y , and let it intersect H O at P . Then P is the midpoint of H O . (1 move)
∙ Draw line M P , and let it intersect line H N at Q . (1 move)
∙ Draw the circle centred at Q , and let it intersect line H N at H and A . Then A is a vertex of the triangle we seek. (It is easy to see that A H = 2 M O , which is what we want.) (1 move)
∙ Draw the circle centered at O with radius O A ; let this circle intersect line l at B and C . (1 move)
∙ Draw lines A B and A C . (2 moves)
The total is 15 moves.