compass and straightedge construction of inverse point


Let c be a circle in the Euclidean planeMathworldPlanetmath with center O and let PO. One can construct the inverse pointMathworldPlanetmath P of P using compass and straightedge.

If Pc, then P=P. Thus, it will be assumed that Pc.

The construction of P depends on whether P is in the interior of c or not. The case that P is in the interior of c will be dealt with first.

  1. 1.

    Draw the ray OP.

    ....OP
  2. 2.

    Determine QOP such that QO and OP¯PQ¯.

    ....OPQ
  3. 3.

    Construct the perpendicular bisectorMathworldPlanetmath of OQ¯ in order to find one point T where it intersects c.

    ....OPQT
  4. 4.

    Draw the ray OT.

    ....OPQT
  5. 5.

    Determine UOP such that UO and OT¯TU¯.

    ....OPQTU
  6. 6.

    Construct the perpendicular bisector of OU¯ in order to find the point where it intersects OP. This is P.

    ....OPQTUP

Now the case in which P is not in the interior of c will be dealt with.

  1. 1.

    Connect O and P with a line segmentMathworldPlanetmath.

    ...OP
  2. 2.

    Construct the perpendicular bisector of OP¯ in order to determine the midpointMathworldPlanetmathPlanetmathPlanetmath M of OP¯.

    ...OPM
  3. 3.

    Draw an arc of the circle with center M and radius OM¯ in order to find one point T where it intersects C. By Thales’ theorem, the angle OTP is a right angleMathworldPlanetmathPlanetmath; however, it does not need to be drawn.

    ...OPMT
  4. 4.

    Drop the perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath from T to OP¯. The point of intersection is P.

    ...OPMTP

A justification for these constructions is supplied in the entry inversion of plane.

If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.

Title compass and straightedge construction of inverse point
Canonical name CompassAndStraightedgeConstructionOfInversePoint
Date of creation 2013-03-22 17:13:17
Last modified on 2013-03-22 17:13:17
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 12
Author Wkbj79 (1863)
Entry type Algorithm
Classification msc 51K99
Classification msc 53A30
Classification msc 51M15