page_title: Mohr–Mascheroni theorem
section_title: Circle center is not collinear with the line
hierarchical_section_title: Mohr–Mascheroni theorem / Constructive proof / Intersection of a line and a circle (construction #4) / Circle center is not collinear with the line
caption_attribution_description: English: Image created in Geometer's Sketchpad. Depicts the construction of the intersection points P,Q between the line defined by AB and the circle c. Mohr-Mascheroni (compass-only Euclidean) construction.
context_page_description: In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.
  It must be understood that by "any geometric construction", we are referring to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed, even though no visual representation of the line will be present. The theorem can be stated more precisely as:
  Any Euclidean construction, insofar as the given and required elements are points, may be completed with the compass alone if it can be completed with both the compass and the straightedge together.
  Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction is functionally unnecessary.
context_section_description: Assume that center of the circle does not lie on the line.
  Given a circle C(r) (in black) and a line AB. We wish to construct the points of intersection, P and Q, between them (if they exist).
  Construct the point D, which is the reflection of point C across line AB. (See above.)
  Under the assumption of this case, C ≠ D.
  Construct a circle D(r) (in red). (See above, compass equivalence.)
  The intersections of circle C(r) and the new red circle are points P and Q.
  If the two circles are tangential then .
  Points P and Q are the intersection points of circle C(r) and the line AB.
  If  then the line is tangential to the circle .
  An alternate construction, using circle inversion can also be given.
  Given a circle C(r) and a line AB. We wish to construct the points of intersection, P and Q, between them (if they exist).
  Invert points A and B in circle C(r) to points A' and B' respectively.
  The line AB is inverted to the circle passing through C, A' and B'. Find the center E of this circle.
  P and Q are the intersection points of circles C(r) and E(C).
caption: Line-circle intersection (non-collinear case)
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