PLANAR MAPPING.
DRAFT COPY ONLY.
10/24/2008.
© Raimond A. Struble.

Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
http://www.infiniteproduct.info/struplan.htm


Send comments and correspondence to: raimondstruble@yahoo.com

Mathematics Review Subject Classification Number: 51M04.

ABSTRACT.

Employing a simple, visual construction involving concurrent cevians of a triangle, we explore some interesting consequences of a synthetic homeomorphism between internal points and external lines of a triangle. The constructions result in a detailed panorama of fascinating correspondents. An unorthodox treatment of edge-lines of a triangle provides for a tangible three-dimensional interpretation of Ceva's theorem . Some similarly-induced three-dimensional phenomena reflecting non-concurrent cevians are explored, and culminates in the unveiling of a related synthetic homeomorphism between internal points and external planes of an octahedron. Reverting back to a two-dimensional enigma, we prove that any three cevians of a triangle are concurrent (exclusive of the centroid concurrence), if and only if rays passing through their feet (two at a time) intersect edge extensions in three collinear points. (One point may be at infinity.) Together with another (even more tantalizing) triangle homeomorphism employing perpendiculars to concurrent cevians, these developments might be expected to add a distinctive new component to synthetic geometry.

A TWENTY-FIRST CENTURY EPISODE IN EUCLIDEAN GEOMETRY.

It is the purpose of this opening Episode to establish the following theorem, and to illustrate graphically some of its more intriguing kinematic consequences for Synthetic Geometry. To paraphrase from reference [2]; I hope the reader will look forward to some relaxing entertainment and enjoy these developments as one would a beautiful piece of music.
THEOREM 1: Given a triangle in the Euclidean plane, there exists a simple one-to-one bi-continuous correspondence between all internal points, excluding the centroid, and all external lines not encountering the triangle. The external line topology results from the employment of polar-coordinates relative to the centroid. The internal topology is the standard Euclidean one, and the correspondence is defined by an elementary, synthetic construction.
PROOF: The proof follows from a straightforward construction employing concurrent cevians.

FIGURE 1.
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Shown in Figure 1 is a triangle ABC with two extended edges, a given concurrent point of three cevians, and rays through two sets of their feet intersecting the extended edges at a and b. (One of a or b may wander off to infinity along an edge line.) Except for the centroid concurrence, these two intersections determine a unique external line passing through a and b. In Figure 2,

FIGURE 2.
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for a given external line intersecting the extended edges at a and b, a test-ray from b passes through the triangle exhibiting two cevian feet. The resulting concurrent cevians determine the third foot, and the intersection a between an extended edge and a ray through two of the feet. It is clear that a suitable rotation (past B) of the ray from b will place a precisely at a, and produce the desired internal concurrent point, the one in Figure 1, of course. The mechanics of these constructions also clearly demonstrate that the one-to-one correspondence is bi-continuous. Indeed, small changes in the location of the internal concurrent point will produce only small rotations and distance changes in the external line, and vice versa. The external intersections may occur along the extended edges below vertex A, and there is only one intersection when the external line is parallel to edge AB or to edge AC. However, only when the concurrent point is the centroid of the triangle will there be no intersections at all.

It should be noted that the selected, extended edges and cevian foot along the remaining edge, represents only one of three possible constructs . Two others are available using the other vertices.

FIGURE 3.
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Each of these three constructs alone exhausts all of the permitted internal points which then determine all the external lines. Moreover, they simultaneously share the same internal configurations. As suggested in Figure 3, the three constructs necessarily result in three collinear intersections (two-at-a-time) determining a common line. The proof of this fact, however, is delayed temporarily, pending further developments concerning non-concurrent cevians (See Corollary 1). (Perhaps the reader may wish to investigate this topic, beginning with [2] Chapter 13, and three special occurrences.) Fortunately, only one construct is actually required in applying these homeomorphisms. So one can proceed forth, with most of the basic phenomena revealed just using Figures 1 and 2. (From a practical standpoint, of course, Figure 3 aids in obtaining the graphs.)

It may be important at this juncture to emphasize that an "external line at infinity" does not exist in the line-space, just as the centroid does not exist in the internal point-space. Likewise, the edges of the triangle (as all lines through the vertices) do not exist in line-space, just as the points of the triangle themselves do not exist internally. We abuse familiar euphemisms such as "tend to" when "move toward" might be more appropriate.

SOME SIMPLE GEOMETRIC IMPLICATIONS.



It is most instructive to associate the internal and external continuous movements inferred by applications of the synthetic constructions of Figures 1 and 2. Such endeavors demonstrate the significance of all the cevian concurrences, not merely the familiar ones encountered in standard Euclidean geometry experiences. Each application, in its own way, offers new challenges requiring proofs, which are not attempted here. Subsequent developments concerning non-concurrent cevians further demonstrate the surprising significance of all triples of cevians, concurrent or otherwise.

I. As an internal point tends to an edge, then the corresponding external line rotates (or tends) toward that edge. Rather intriguing is the fact that an internal point moving along a smooth curve transversely past the centroid corresponds to a continually turning external line with its angular rotation (mainly) reflecting arc-length changes along the internal curve. In particular, an internal point moving along a cevian corresponds to an external line simply rotating about a center located on the extension of an edge.

FIGURE 4.
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There seems to be no traditional treatments which would substantiate a proof of the configuration claimed for Figure 4. This implication then becomes a new outgrowth of Theorem 1, and the reader is invited to formulate and prove the inferred corollary in the context of traditional Euclidean geometry. Such is the situation for all the items in I through VI. Movements along three concurrent cevians correspond to three rotation centers of lines located at the three intersection points as displayed in Figure 3 (a corollary of a corollary). Lines rotating about other centers correspond to movements of internal points along continuous curves between two vertices.

II. An internal point moving along another line across the triangle enclosing two vertices away from the centroid, corresponds to a continually turning tangent depicting a bulge from one edge to another edge. The configuration is reflective of an ice-cream cone from the opposing vertex. (It would be of interest to further identify such curves.)

FIGURE 5.
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Any three crossing lines, each enclosing two vertices away from the centroid (more-or-less parallel to the edges), correspond to three smooth bulges forming a would-be three-leaf clover (each leaf resembling Figure 5).

III. In other cases where the internal line encloses only one vertex away from the centroid, the continually turning external tangents depict portions of hyperbola-like curves, away from the triangle, with asymptotes as two edge lines. The reader may wish to investigate the actual nature of these curves, which resemble boomerangs.

FIGURE 6.
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The medial triangle, for example, leads to three versions of Figure 6 displayed at the three vertices. Other crossing lines lying outside of the medial triangle correspond to similar boomerangs, but those cutting through the medial triangle correspond to portions of two boomerangs at two vertices (connected by a common tangent, which changes the direction of rotation).

The amazing changes in these three graphs can be partially appreciated if one first examines the three ranges of the angle rotations and how these ranges overlap . The range for the graph in Figure 6 measures the least followed by that of Figure 4 (less than π) and that of Figure 5 (greater than π). Then when placed side-by-side, the progression seems plausible (See VI for further details.)

IV. Also intriguing is the fact that a continually rotating tangent along an external curve (avoiding the triangle) corresponds to the movement of an internal point along some continuous curve. The latter may, however, exhibit sharp corners in the directions of some vertices. The internal arcs corresponding to a circumcircle, for example, do not connect with one another internally and form spikes at the vertices.

V. An interesting fact is that a continuously departing, non-rotating external line parallel to an edge, corresponds to an internal point moving along a median to the centroid. As the external line returns from "infinity " to the apposing vertex, the corresponding internal point completes the median. With other orientations of parallel departing lines, however, the internal point moves along a smooth curve from the contacting vertex bending into the centroid.

VI. The most startling behavior in the external line-space consists of a disjoint shifting of the location of rotation, which nonetheless results in a continuous external motion. Here a single external line participates in the shift across the external "Euclidean" landscape. For example, an external polygonal curve, with rotations filling in the external corners (while avoiding the triangle), corresponds to internal smooth arcs connecting internal corners, like a game of ping-pong, where one participant slams shots while the other returns shots using lots of English. If Figures 5 and 6 are superimposed, then the two external curves share a single tangent line corresponding to the point of intersection of the two lines across the triangle. At the external points of tangencies, the directions of motions differ, and the two curves, as viewed, are completed by further movements in the two directions. However, if the internal movement changes, so that a continuous motion executes a turn and forms a corner at the intersection point, then the corresponding continuous external movement changes direction and produces a shift along the common tangent line between a portion of the bulge and a portion of the boomerang. Should we now adjoin a cevian, such as in Figure 4, which passes through the internal intersection of the crossing lines, the continuous movements internally become multiple, yet rather simply envisioned, while the corresponding external movements (including rotations about a center located between the external points of tangencies) also become multiple, yet somewhat challenging to depict. The latter may be observed as partially smooth motions in various directions and partially rotary motions. All changes (shifts) take place across the single common external tangent line which corresponds to the internal concurrent point of the three crossing lines.

THE TOPOLOGICAL VIEW.

It is easy to see that the Euclidean plane, minus the origin, is homeomorphic to the topological line-space consisting of ALL lines which do not pass through the origin. A simple synthetic, origin-centered homeomorphism assigns to each point P the line through P perpendicular to the ray from P to the origin. Therefore, the subspace of lines which do not encounter a triangle, and whose centroid lies at the origin, is necessarily homeomorphic to some Euclidean subspace of the plane. Figure 1 identifies such a Euclidean subspace as the punctured interior of the triangle and provides for a synthetic homeomorphism. Although these two synthetic homeomorphisms differ markedly from one another in the interiors of triangles, the topology of their exterior line-spaces is the same. So it seems appropriate to briefly examine some manifestations of the origin-centered homeomorphism. Generally, one finds simple, routine relationships between smooth point-curves and continuous smooth tangent-induced curves. Relative to the origin, both can be convex, either one can be concave, while the other is convex, but not both can be concave. Whenever they make contact, they share a line at the point of contact. A straight line of points corresponds to an arc of a tangent-induced curve resembling a parabola with focus at the origin. On the other hand, a center of rotating lines corresponds to a circle of points passing through the rotation center and the origin. The reader may wish to examine other interesting manifestations of this homeomorphism, such as the tangent-induced loop corresponding to a point moving along the parabola. Fortunately, this homeomorphism is quite useful in further exposing the synthetic homeomorphisms of Figure 1 .

When each (and every) triangle is located so that its centroid lies at the origin, then the intriguing consequences of the various homeomorphic mappings of the Figure 1 become displayed by lines lying in the Euclidean plane, accompanied by a common topology for lines. Variations resulting from modifications of the triangles are then conveniently pursued within one (and the same) topological line-space. It is helpful to begin considerations employing a Euclidean plane, minus the origin, and the topological line-space consisting of those lines that avoid the origin. Then upon placing a triangle with its centroid at the origin, one simply disposes of those lines which encounter the triangle, and reinstates the interior of the triangle as a Euclidean point-set which is homeomorphic to what is left of the lines. According to the origin-centered homeomorphism, the disposed line-space is homeomorphic to the Euclidean point-set with a border defined by three circular arcs passing through the vertices and meeting along the edges. (The completed circles convert the centroid into a Miguel point [2].)

Of course, the synthetic one-to-one constructions of Figures 1 and 2 can be pursued rather fruitfully without any topological concerns, if so desired. However, the obvious kinematic linkage in Figure 1 , induced by continuity considerations, provides a very dynamic picture for various real-life applications, say, in engineering mechanics, electronics, optics, etc. This dynamic picture is further enhanced by reference to Figure 3 . It is certainly an enigma to recognize that this rather special, elementary 10-line configuration has apparently been overlooked. (In this connection, see Figure 3.4A of reference [1].) It is fascinating to observe that the complete 10-line configuration persists even in the extreme situation where one triangle vertex recedes to infinity. In Figure 12 , an alternative, 10-line configuration only adds to the historical enigma.

APPENDIX 1.
TRANSFORMATION FROM WITHIN A TRIANGLE.

It may have occurred to the reader that the original correspondence of Figure 1 is, in effect, a continuous mapping from most (not all) internal concurrent points to pairs of points lying on two extended edges. One can use the two selected edges as coordinate axes for points in the plane, so that the two intersections, a and b, then become the coordinates of an external point (particularly, when the base vertex is right-angled and the coordinates become the customary Cartesian coordinates). This transformation is much less startling than the one considered in the Episode, but more startling than what is known as inversion of a circle. It clearly deserves a vigorous pursuit, and could be labeled TRANSFORMATION FROM WITHIN A TRIANGLE. As with the central point for inversion of a circle, the troublesome points here, lying along two medians dividing the interior region into four components, might be viewed as collectively sent to infinity. Unlike the circle situation, for each triangle there are three such transformations available (one from each vertex , where the four internal components differ). Within each component, the transformation is topological (in the customary sense for points in the Euclidean plane).

NON-CONCURRING CEVIANS.

It may also have occurred to the reader that when the three constructs of Figure 3 are applied to the feet of arbitrary cevians, as in Figure 7,

FIGURE 7.
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this construction establishes a continuous mapping from most (not all) triples α    β    γ of points on the triangle edges to triples of points a    b    c on the edge extensions. The troublesome triples α    β     γ, of course, are those not producing three finite intersections a, b and c with the extended edges. (They are readily identified.) So one can artificially extend the mapping by sending troublesome triples collectively to infinity and then, more-or-less, ignoring them in some of the following procedures, where only finite triples a    b    c are inferred. On the other hand, one should note that the discovery technique of Figure 2 can be modified so as to obtain the triangle triple α    β    γ in Figure 7 corresponding to any given external triple a    b    c, even if one or two of a, b and c wanders off along an edge line to infinity. For example, a test-ray from point b yields tentative β and γ intersections. Then a ray from point c to γ yields a tentative α intersection. The ray through β and α then yields some intersection point a along CB extended. The initial test-ray from b can always be rotated so that the intersection a becomes any given point a along CB extended.

An interesting view of arbitrary (non-concurring) cevians blossoms by associating with each such internal triangle of Figure 7 , the external triangle abc whose edges never encounter the primary triangle ABC . In this view, one observes three telescoping triangles which are topologically related through the synthetic construction indicated.

If the edge-lines of Figure 7 are separately slid (rigidly) along the medians to the triangle centroid, as in Figure 8,

FIGURE 8.
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then the mapping can be re-interpreted from the translated triples α1    β1    γ1 to the translated triples a1    b1    c1. However, the translated edge-lines can also be re-interpreted as coordinate axes in THREE-SPACE, with p=(α1,β1,γ1) and P=(a1,b1,c1) representing the coordinate expressions of the corresponding three-dimensional points. These three-dimensional points lie inside and outside of a three-dimensional, slanted box (parallelopiped), centered at the origin with apposing quadrangular faces passing through D and E, F and G, H and I, of Figure 9. For an isosceles right-triangle, the three-dimensional coordinates are customary Cartesian ones, and the box is rectilinear.

FIGURE 9.
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The eight corners of the box are not displayed in Figure 9 , but decidedly need to be visualized, as in Figure 10 . They can be identified using triples α    β    γ lying variously about the vertices of triangle ABC, and whose locations then determine the numerical signs of their three-dimensional coordinates. The actual interior points p = (α1, β1, γ1) encountered necessarily lie within the octahedron DEFGHI whose faces appose the eight corners of the box. The corresponding exterior points P = (a1,b1,c1) necessarily lie outside of the box. Consequently this continuous mapping from most inside points within the octahedron to points outside of the box could now be labeled TRANSFORMATION FROM WITHIN AN OCTAHEDRON.

FIGURE 10.
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Aside from the transformation from within a box, the above sliding technique (Figure 8) might be of some special interest. As an instructive example, we recall that according to Ceva's Theorem, the concurrence of the three cevians in Figure 11 is equivalent to the numerical equality L1 × L3 × L5 = L2 × L4 × L6, of these triple products.

FIGURE 11.
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When re-interpreted, as in Figures 9 and 10, this equality is equivalent to the equality of the volumes of the two corner sub-boxes meeting at a point inside the octahedron. Indeed, the three dimensions of the two sub-boxes are the triples of lengths L1, L3, L5 and L2, L4, L6, just as those of the primary box are L1 + L2, L3 + L4, L5 + L6. In this setting, the two sub-boxes fill the diagonally opposing corners in the primary box, and expand so as to share an inside corner point for each (determined from the triple α    β    γ). The movement of this common corner point clearly dictates the relative volumes (sizes) of the two connected sub-boxes, so that Ceva's criterion certainly becomes tangible. These two sub-boxes happen to possess the same set of dimensions for each of the centroid and Gergonne concurrences. (See [2] Chapter 1, 7 and 12 for other tricks which could be applied to pairs of sub-boxes with matching volumes. The first theorem in Chapter 12, for example, reflects a simple interchange of the two sub-boxes.)

We note that the subset of points within the octahedron stemming from concurrent cevians becomes a two-dimensional submanifold, which intersects each ray parallel to a coordinate axis in exactly one point. For with β and γ held fixed in Figure 7, a single location for α along AC determines the feet of three concurrent cevians. Similarly, with just α held fixed in Figure 7, the pairs β γ producing three concurrent cevians along αB, depict in Figure 10, the intersection of the submanifold as a simple continuous curve lying on the associated plane-slice across the octahedron, and which connects two apposing edges of the octahedron. Since this special occurrence transpires simultaneously for slices in each of three orientations, the submanifold is seen as a smooth curtain issuing from a closed circuit consisting of six edges of the octahedron (avoiding those defining the positive and negative faces) and passing through the origin.

We would be negligent not to observe that the (extended) mapping from all non-origin inside points p = (α1, β1, γ1) of the octahedron to planes passing through the external intercepts a1, b1 and c1 of Figure 8, yields a homeomorphism in three-space that clearly extends that of Figure 1.

In three-space, the planes a1b1c1 intersect the axes at points (including infinity) exterior to the extremes D, E, F, G, H and I of Figure 10, and so lie outside the octahedron. No exterior plane corresponds to the origin. In fact, upon bypassing the "slices" of Figure 8, Figure 7 tells the whole story synthetically: in two-space for concurring cevians, triangle abc collapses into a single external line (Figure 3); in three-space, triangle abc spawns a space triangle for all triples of cevians, concurrent or not (even if one or two of the intercepts a, b or c departs to infinity). Therefore, it seems appropriate to summarize this view with a formal statement of these observations as our second theorem:
THEOREM 2. The three-dimensional interpretation of Figure 8 produces a synthetic homeomorphism which maps the (punctured) inside of the octahedron DEFGHI, as in Figure 10, onto the space of all planes not encountering the octahedron. This homeomorphism is defined by the associated elementary synthetic construction illustrated in Figure 7.

TWO SIGNIFICANT COROLLARIES.

Figure 7 now leads us to the delayed proofs of our two primary corollaries of Theorem 1.
COROLLARY 1. The three external intersections of Figure 3 are collinear, so long as the three internal cevians are concurrent at other than the centroid.

PROOF. For any three non-concurrent cevians, as in Figure 7, a small internal triangle is produced, and for which its three (non-centroid) vertices correspond, according to Theorem 1, to three distinct external lines. If the small internal triangle is continuously merged into a concurrency condition anywhere except at the centroid, then the three external lines must necessarily merge into one, and the same, single line corresponding to the triple concurrency point. Because of the persistent single concurrencies at each of the vertices of the small triangles, producing rays βγb, γαc and αβa, the three external intersections a, b and c necessarily move onto this single line, and are collinear.
The converse follows immediately when the discovery technique in Figure 2 is applied to a couple of the external intersections. For then, Corollary 1, applied to the resulting triple concurrency point, incorporates the third external intersection corresponding to the three concurring cevians.
COROLLARY 2. The three internal cevians of Figure 3 are concurrent, so long as the three external intersections are collinear.
The reader is invited to supply direct, non-topological proofs of these corollaries.

ANOTHER TWO-DIMENSIONAL SYNTHETIC HOMEOMORPHISM.

The reader is also invited to establish the following (apparently overlooked) synthetic construction.
THEOREM 3. Given a triangle in the Euclidean plane, the perpendiculars to three concurrent cevians, passing through a point of concurrency other than the orthocenter, intersect edge extensions in three collinear points (One point may be at infinity.)
FIGURE 12.
More frequently than not, however, the intersections lie along a single edge-line, and they may include vertices. The transition from these edge-lines to external lines must take place along circular arcs determined by right-angled legs passing through two vertices at a time. For obtuse or right-triangles, some external lines are missing. However, for ACUTE triangles, all external lines emerge, and the construction, restricted to the punctured region bounded by the three circular arcs centered at the mid-points of the edges, becomes another synthetic homeomorphism onto the space of lines not encountering the triangle. (The reader may wish to prove that these circular arcs meet along the triangle edges, as displayed.) In this setting, therefore, the construction exhibits analogous graphical correspondences. For example, points lying along the truncated altitudes correspond to external lines parallel to the edges. Since two intersections alone determine each external line, the proof of the analogous working theorem readily follows (as that of Theorem 1) when the internal concurrent points employed are limited to this special punctured region (excluding the orthocenter). The reader may wish to explore the analogous graphs using this alternative "perpendicular rule" construction. This will lead to a second Twenty-first century Episode in Euclidean Geometry.

REFERENCES.

1. Coxeter HSM, Greitzer SL.
Geometry Revisited.
New Mathematical Library.
Washington, DC: Math Assoc Amer. 1967.
ISBN: 0883856190, 207 pages.

2. Honsberger R.
Episodes in Nineteenth and Twentieth Century Euclidean Geometry.
New Mathematical Library. Washington DC: Math Assn America. 1996. Second printing. 2005.
ISBN: 0883856395, 174 pages.


Last updated: 10/24/2008, by Raimond A. Struble, PhD.