Difference between revisions of "Construction"

Latest revision as of 19:52, 28 August 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. Hence, construct a parallel line to line $l$ through point $A$ not on $l$ .

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 $n$ different parts.

8. Construct length $ab$ given lengths $a$ and $b$ and unit segment $1$ .

9. Construct $a/b$ and $\sqrt{ab}$ . Hence, construct $\sqrt{a}$ given unit segment $1$ .

10. Construct a tangent to a circle.

11. Construct common tangents to two circles.

12. Construct a parallelogram with side lengths a and b. Hence, construct a square with side length a.

These basic constructions should be easy to accomplish. Now, try these:

13. Construct a line passing through a point $P$ parallel to line $l$ .

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 $P'$ on ray $CP$ such that $CP * CP'$ equals the square of the radius of C. Hence or otherwise, construct the inverse of a point P using compasses only.

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.

21. Construct $sin C, cos C, tan C$ given unit segment $1$ and acute angle $C$ .

22. Construct a right triangle with the given lengths of a hypotenuse and altitude to the hypotenuse.

23. Construct $15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ$ angles. Hence or otherwise, construct a right triangle whose median to the hypotenuse is equal to the geometric mean of the legs. (Source: IMO)

Good luck!

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Difference between revisions of "Construction"

Latest revision as of 19:52, 28 August 2014

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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.
-{{stub}}
+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:
+. Duplicate a line segment.
+. Copy an angle. Hence, construct a parallel line to line <math>l</math> through point <math>A</math> not on <math>l</math>.
+. Construct an angle bisector.
+. Construct a perpendicular bisector.
+. Construct a perpendicular from a point to a line.
+. Construct a triangle with side lengths a, b, and c.
+. Partition a line segment into <math>n</math> different parts.
+. Construct length <math>ab</math> given lengths <math>a</math> and <math>b</math> and unit segment <math>1</math>.
+. Construct <math>a/b</math> and <math>\sqrt{ab}</math>. Hence, construct <math>\sqrt{a}</math> given unit segment <math>1</math>.
+. Construct a tangent to a circle.
+. Construct common tangents to two circles.
+. Construct a parallelogram with side lengths a and b. Hence, construct a square with side length a.
+These basic constructions should be easy to accomplish.
+Now, try these:
+. Construct a line passing through a point <math>P</math> parallel to line <math>l</math>.
+. Construct a square circumscribed on a circle.
+. Construct a regular hexagon inside a given circle.
+. 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. Hence or otherwise, construct the inverse of a point P ''using compasses only''.
+. Construct a square, all of whose vertices are on a given triangle.
+. Construct a regular pentagon.
+. Construct the [[radical axis]] of two circles.
+. 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.
+. Construct <math>sin C, cos C, tan C</math> given unit segment <math>1</math> and acute angle <math>C</math>.
+. Construct a right triangle with the given lengths of a hypotenuse and altitude to the hypotenuse.
+. Construct <math>15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ</math> angles. Hence or otherwise, construct a right triangle whose median to the hypotenuse is equal to the geometric mean of the legs. ''(Source: IMO)''
+Good luck!
 [[Category:Definition]]
 [[Category:Geometry]]