Difference between revisions of "Construction"
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'''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 [[axiom]]s to create various figures in the [[Euclid]]ean plane. | '''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 [[axiom]]s to create various figures in the [[Euclid]]ean 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 <math>n</math> different parts. | ||
| + | 8. Construct length <math>ab</math> given lengths <math>a</math> and <math>b</math>. | ||
| + | 9. Construct <math>a/b</math> and <math>\sqrt{ab}</math>. | ||
| + | 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 <math>P</math> parallel to line <math>l</math>. | ||
| + | 14. Construct a square circumscribed on a circle. | ||
| + | 15. Construct a regular hexagon inside a given circle. | ||
| + | 16. Construct the [[Inversion|inverse]] of a point P with respect to circle 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! | ||
[[Category:Definition]] | [[Category:Definition]] | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
Revision as of 13:32, 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.
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!
