HCR’s Theorem & Corollary Copyright©3-D Geometry by H. C. Rajpoot HCR’s Theorem Rotation of two coplanar planes, meeting at angle-bisector, about their intersecting edges     Master of Technology, IIT Delhi Introduction: This theorem says that if two coplanar planes (i.e. lying in the same plane), meeting each other at a straight edge which is bisector of angle between their intersecting straight edges, are to be rotated through same angle about their intersecting straight edges then it is first required to cut remove V-shaped plane symmetrically about their common straight edge & then planes are rotated about their intersecting straight edges through a desired angle. But if these two coplanar planes have to be rotated through a desired angle about their intersecting straight edges such that their new edges (generated after removing V-shaped planar region) coincide each other then we require a specific angle (i.e. V-cut angle) to cut remove V-shaped plane to allow rotation of the co-planar planes meeting at a common edge. In this theorem, we have to derive a mathematical expression to analytically compute the V-cut angle () required for rotating through the same angle () the two co-planar planes, initially meeting at a common edge bisecting the angle () between their intersecting straight edges, about their intersecting straight edges until their new straight edges (generated after removing V-shaped planar region) coincide. As a result, we get a point (apex) where three planes intersect one another out of which two are original planes (rotated) & third one is their co-plane (fixed). This theorem is very important for creating pyramidal flat containers with polygonal (regular or irregular) base, closed right pyramids & polyhedrons having two regular n-gonal & 2n congruent trapezoidal faces. HCR’s Theorem: If two co-planar planes initially meet or intersect each other at a straight line (edge) which bisects the angle  (  ) between two intersecting straight edges of the planes then V-cut angle , required to cut remove V-shaped plane bisected by the common edge so that two planes (after cutting V- plane) are rotated through the same angle  about their intersecting straight edges until their new edges (i.e. generated after cut-removing V-plane) coincide, is given by following formula     (      )      (        )  Where,   is the dihedral angle between rotated cut planes when their new edges coincide such that            Proof: Consider two planes 1 & 2 initially lying in the same plane (i.e. plane of paper) such that they meet or intersect each other at a common straight edge AB which bisects angle  (  ) between straight edges BC & BD intersecting each other at point B (as shown in the figure-1). It is to cut remove V-plane to allow rotation Now, in order to cut remove V-shaped plane equally divided by common edge AB, we make V-cut angle  bisected by common edge AB. Mark the V-shaped planar region (as shaded) which is to be cut removed so as to rotate the planes 1 & 2 through the same angle until their new edges coincide (See fig-2 below) Figure 1: Two co-planar planes meet at edge AB & their edges BC & BD intersect each other at angle  which is bisected by common edge AB. The plane of paper is taken as co-plane in which two planes 1 & 2 initially lie