CONCURRENT CEVIANS.
© 2004-2009. RAIMOND A. STRUBLE, PhD.
DRAFT COPY ONLY.
9/27/2009.
© Raimond A. Struble.
Send comments and correspondence to: Raimond A. Struble,
P. O. Box 50376, Raleigh, NC 27650-6376
and emails to: George.Moore4@va.gov
Prove that if three cevians of a triangle are concurrent (at points other
than the centroid), then extensions of the three lines passing through their
feet intersect extensions of the edges of the triangle along an external
line. Conversely, when extensions of the edges of a triangle intersect
at three points of an external line, then there exist concurrent cevians
possessing feet which are collinear with those intersections. (Appropriate
versions hold for interesections at infinity, so long as the centroid
is expunged.)
FIGURE 1.
2551.
Upon defining a suitable topology for the space of external lines, show that
the one-to-one correspondence between internal (concurrence) points
and external lines becomes topological.
Submitter's comments:
This problem is a bit of an extensive (but delayed) project in synthetic
geometry, with a suitable metric distance, for example, being defined for two
external lines as the sum of the (acute) angle they form, and the difference
between their Euclidean distances from the centroid. Proofs are very
challenging, but implications are easy and rather intriguing,
as illustrated below.
SOME IMPLICATIONS.
For example, as an internal point tends to an edge, then the corresponding
external line rotates toward that edge. Most intriguing is the fact that an
internal point moving along a smooth curve (not near the centroid)
corresponds to a continually turning external line with its angular rotation
(mainly) reflecting arc-length changes along the 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 2.
2552.
Three of these centers are always collinear for three concurrent cevians,
and are located (as originally claimed) on intersections between edge-line
extensions and lines passing through their feet. An internal point moving
along another line across the triangle (avoiding the vertices and 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.
FIGURE 3.
2553.
The corresponding configuration stemming from the medial triangle consists of
three bulges forming a would-be three-leaf clover. A similar configuration
stems from the internal portions of the three mutually tangent circles
centered at the vertices, as well as the incircle.
Also very 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. In particular, the tangents to
a closed (say, convex) curve surrounding the triangle correspond to a closed,
continuous, internal curve. However, the internal arcs corresponding to the
circumcircle do not connect with one another internally,
but form spikes at the vertices.
(See the discussion concerning FIGURE 4 for an explanation.)
These two major intriguing "facts" come together very nicely for a circle
centered externally and passing through the triangle. An outer portion
of this circle provides for tangents avoiding the triangle and corresponds
to an internal,
boomerang-shaped
portion, turning away from the circle. Then
two separate internal curves, extending from vertices to mid-points of the
edges, provide for continually-turning tangents to concave faring-curves
which connect (in the Euclidean sense) the circle with two extended edges,
at infinity. The observed
vertical segments coming from the vertices
result from the external lines
tending to vertices. The Euclidean limit does NOT then represent
a line-space limit, i.e., lines through the vertices do not exist in
line-space, just as triangle points do not exist in the internal
Euclidean subspace. Likewise, the centroid does not exist internally,
just as infinity does not exist externally. (Topologically speaking,
these spaces are open rings.)
FIGURE 4.
2554.
Other interesting scenarios are certainly possible,
say, by halting the rotation of the external tangent lines
short of the vertices, so that the resulting internal curves
connect with the central portion,
internally, forming spikes (due to the changes in curvature
of the external configuration).
Reversing the situation a little, we note that a point moving along
an internal subtriangle (not surrounding the centroid) corresponds to
a continually-turning external line depicting tangents to a connected curve
with spikes. Indeed, the edges of the internal subtriangle produce three
portions of this curve, but where the acute vertices force changes in the
movement along the portions. In the general situation (i.e., no edge of the
subtriangle points to a vertex of the primary triangle), this leads to
a configuration with two spikes for an obtuse subtriangle and to three
spikes for an acute subtriangle.
FIGURE 5.
2555.
If an edge of the subtriangle points to a vertex, then a spike collapses
into a rotational center.
An interesting fact is that a continuously departing, non-rotating external
line, parallel to an edge, corresponds to an internal point moving
continuously along the meridian from that edge to the centroid. With other
orientations, however, the internal point moves along a ray to the centroid
from an internal point.
All the above facts concern continuous operations of interest within the
open, interior of the triangle (sans the centroid) and the open exterior
lines not contacting the triangle. To illustrate an interesting fact directly
concerning synthetic geometry, we note that any three internal points
(not surrounding the centroid) correspond to an external triangle,
whose Euler-line, in turn, corresponds to an internal Euler-point
(my definition) falling within the triangle with these three initial points
as vertices. The relationships between such an Euler-point and its triangle
certainly could use some sort of clarification.
As noted, if we begin with the total subtriangle, which leads to a spiked
configuration, then (in general) the internal Euler-line of the subtriangle
yields a connected curve, lying inside of the configuration. This becomes
tangent to the configuration at two external Euler-points
(my definition) corresponding to the intersections of the edges of the
subtriangle with the extension of the Euler-line of the subtriangle.
These Euler-points would seem to merit some sort of recognition.
If the Euler-line of the subtriangle happens to point to a vertex of the
primary triangle, then the external Euler-points can be located using
two particular rays of a rotation center, which are tangent to the spiked
configuration. Incidentally, the Euler-line of the primary triangle
(passing through the centroid) produces two separate external curves
swinging away from two edges to infinity.
Many other, similar concurrence situations of standard synthetic geometry
could also use some clarification.
Here the entire synthetic Euclidean geometry internal to a triangle
is manifested in a new topological geometry of external line-space.
It doesn't seem to matter much which primary triangle is employed,
although the picture variations occurring with the morphing of the triangles
may be rather interesting. Perhaps standard synthetic geometry could use
other intriguing reinterpretations on the basis of this topological
transformation, following an uncovering of a proof of the proposed problem.
APPENDIX. CONCURRENT CEVIANS.
DIFFERENTIAL CALCULUS.
Consider a smooth internal "arc", and the corresponding smooth external
"Arc", induced by the continually turning external lines, which correspond
to an internal point moving along the "arc".
FIGURE 6.
2556.
If p and q are neighboring points along "arc", reflecting
a secant line passing through these points, then the corresponding external
lines P and Q are tangent to "Arc", reflecting a crossing-point
between the contact-points. As q tends to p and the secant
tends to the internal tangent line T, the "arc" at p, then
the external tangent line Q tends to P and the crossing-point
slides along P to the contact-point t of the "Arc". Therefore,
just as the internal point p corresponds to P, the external
contact-point t corresponds to the internal tangent T.
Conversely, if t is the contact point of the tangent line P
with the external "Arc", then the same procedure, using a second tangent line
Q, leads to internal secant lines connecting points p and
q along the internal "arc" whjich tend to the tangent T
at p. When this is done for various points t lying along the
external "Arc", one obtains various tangents P(t) externally and
various points p(t) internally. Simultaneously, one obtains internal
tangents T(t) with the internal "arc" at the points p(t).
FIGURE 7.
2557.
Having selected an external "Arc",
P(t)
becomes a solution curve of the derivative equation:
slope of p(t) at t = slope of the specified T(p(t)
Illustrative examples might help to clarify the situation. Suppose that the
extenral "Arc" binds slightly inward toward the centroid. Then following the
bend, the contact points t necessarily move toward the intersections
of the tangents P(t) and the perpendiculars to the centroid. If for
example, the "Arc" is a portion of the ice-cream surface of an ice-cream
cone, then p(t) results in a straight line segment (not pointing
at a vertex), and T(t) = the line segment. This fulfills the
requirements of D.E. Incidentally, the "parameter" t varies
consistently and constantly throughout the various correspondences.
On the other hand, if the internal "arc" is selected, then the corresponding
tangent line along the external "Arc" are detemined, as well as the "Arc"
itself and the contact points. Suppose that one selects the vertex-to-vertex
internal "arc" of Figure 4. Then the external "Arc", of course,
is the outer portion of the circle in this figure. For each contact point
t along the circle, an external tangent line P(t) is obtained,
along with the internal point p(t) somewhere along the internal
vertex-to-vertex curve. The internal tangent T(t) at p(t)
is obtained using secants to neighboring points q, confirming that the
contact point t is the limit of the crossing point between Q
and P(t). So D.E. is again fulfilled.
All-in-all, this calculus link between the external line space
and the (punctured) interior Euclidean space suggests that new closed-form
antiderivatives might somehow be constructed.
APPENDIX. CONCURRENT CEVIANS.
DIFFERENCE EQUATIONS.
There are other aspects of Figure 6 that can be put to good use.
One amounts to a discrete analogue of the above derivative equation
theory. For upon using a selection of a finite number of closely-spaced
points lying along an internal "arc", the corresponding external lines will
form a discrete set of slightly rotating tangents to an external "Arc".
The latter is viewed, of course, as an unknown limiting image, but the finite
number of contact points can be well-approximated using the graph.
Conversely (so to speak), for a precisely selected external "Arc",
any discrete set of rotating tangents at precisely selected points
along the "Arc" will correspond to a finite set of interior points
lying precisely along the associated interior "arc",
with secant slopes specified by the geometry.
The latter process
affords tremendous promise for finding exact solutions to previously
undreamt-of difference equations. Whenever the slopes of the secant
segments are specified, difference equations emerge.
Last updated: 9/27/2009, by Raimond A. Struble, PhD.