Difference between revisions of "Construction"
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15. Construct a regular hexagon inside a given circle. | 15. Construct a regular hexagon inside a given circle. | ||
| − | 16. Construct the | + | 16. Construct the inverse of a point P with respect to circle C. In other words, construct the unique point <math>P'</math> on ray <math>CP</math> such that <math>CP * CP'</math> equals the square of the radius of C. |
17. Construct a square, all of whose vertices are on a given triangle. | 17. Construct a square, all of whose vertices are on a given triangle. | ||
Revision as of 12:35, 15 June 2014
Constructions with straight edge and compass (i.e. the ability to mark off segments, draw circles and arcs, and draw straight lines) are a branch of geometry that rely on the use of basic geometrical axioms to create various figures in the Euclidean plane.
A compass is a tool that can draw circles and arcs of circles.
A straightedge is an unmarked ruler that can draw line segments.
No other tools are allowed in a construction. However, the two basic tools alone can allow one to:
1. Duplicate a line segment.
2. Copy an angle.
3. Construct an angle bisector.
4. Construct a perpendicular bisector.
5. Construct a perpendicular from a point to a line.
6. Construct a triangle with side lengths a, b, and c.
7. Partition a line segment into 
different parts.
8. Construct length 
given lengths 
and 
.
9. Construct 
and 
.
10. Construct a tangent to a circle.
11. Construct a common tangents to two circles.
12. Construct a parallelogram with side lengths a and b.
These basic constructions should be easy to accomplish. Now, try these:
13. Construct a line passing through a point 
parallel to line 
.
14. Construct a square circumscribed on a circle.
15. Construct a regular hexagon inside a given circle.
16. Construct the inverse of a point P with respect to circle C. In other words, construct the unique point 
on ray 
such that 
equals the square of the radius of C.
17. Construct a square, all of whose vertices are on a given triangle.
18. Construct a regular pentagon.
19. Construct the radical axis of two circles.
20. Given two chords of a circle intersecting in the interior of the circle, construct another circle tangent to the chords and internally tangent to the original circle.
Good luck!
