I hope you had fun with the game, just like I do, the answer provided here are meant to be a sample solution. There are, of course, other ways to accomplish the same goal. I am sure the answers are correct, but in case you think my answer is inaccurate or you have better solutions, please leave me comments!
Andrew's Euclid Solution
Monday, June 23, 2014
Level 20: Construct a line (segment) tangent to both circles. Construct the outer tangent line.
Hint: Think ratios.
Solution: First construct the ray AB, and
let the outer tangent line intersect AB at O. Now OA:OB should be (radius of A:
radius of B). To find such point O, we use the similar triangle NAO and MBO.
Once we found the point O, we need to find a point C on circle A such that OCA
is a right angle. Now we leverage another fact, angle subtended by the diameter
on a circle is the right angle, so we construct the midpoint P of O and A, draw
the circle with center P and radius PA, so the intersection between the circle
would give us the point C and we are done.
Level 19: Construct two points, such that the segment is cut into three equal pieces.
Hint: Perhaps unrelated – study the
centroid of a triangle.
Solution: The centroid of a triangle is
found by intersecting the median, the median is defined by the intersection
between the midpoint of an edge and the opposite vertex. The interesting thing
about this is that it can be shown that the intersection point always cut the
median into two-third on the vertex side and one third on the remaining side.
It is this property that we will leverage for this level.
With
that in mind, we make AB the median, which means A is the opposite vertex and B
is the midpoint of some base line, construct the base line by creating a small
circle at B and draw the diameter CD, find the midpoint E of AD and join CE for
another median, now we find F which is the centroid of the triangle ACD. That cuts
the line in two third/one third, and so we use another circle with center F and
radius FB to find the remaining point G.
Level 17: Given a point A, a line, and a point B. Construct a circle that passes through A and is tangent to the line at B.
Hint: The circle passes through A and
B, what else?
Solution: The circle must also pass through
the mirror image of A. The line is a tangent, which means the line through B
perpendicular to it is the diameter. The point D, the mirror image of A, can be
constructed by first constructing a parallel line through A parallel to the
tangent, and intersect it with the circle with center B and radius AB. The
rest? It is just level-15, the circumcircle of A, B and D!
For
economy we could have just constructed the perpendicular bisector of AB and
then the line through B perpendicular to the tangent and use that as the
circumcenter, but it would be hard to explain why that’s true.
Level 16: Given a line, a line segment CD, and a point O. Construct a circle with center O that cuts off a segment congruent to CD on the given line.
Hint: The same
old hint – perpendicular bisector of chord pass through center.
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