Final Write Up
††††††††††† Summer 03
Shown below is an arbitrary acute triangle with an arbitrary interior point.† The sides are constructed of extended lines which are hidden for clarity.† The construction was done on the extended lines to allow for exploration of the point P outside of the triangle.†
Segments from a vertex to the arbitrary point is shown.† A relationship appears to emerge between the product of the lengths of the three side segments shown in bold (AF, BD and CE) and the product of those shown in dashed lines (FB, DC, and EA).† The relationship is that these products appear to be equal as shown in the measurements on the figure above.† Although this appears to be a clear relationship, that is yet to be proved.
A hint was given to consider parallel lines and similar triangles as a strategy for formulating a proof.† To engage this hint, start with a line parallel to the base of the triangle through the vertex A:
that one would like to establish relationships among the segments within this
triangle associated with AF, FB, BD, DC, CE and EA.† So it would appear to be beneficial to unhide the extended lines through
point P in search of these similarities.
A number of similarities containing the segments of interest are readily observable.† From knowing the equality of interior angles of a line intersecting parallel lines and from knowing the equality of angles opposite the intersection of two line, the following observations are made:
Noting these equality of these pairs of angles, several pair of similar triangles can be identified.† To find relationships among the various segment, one may suppose that including a segment of interest along with a segment common to other similar triangle into a set of similar triangles may be useful.† For example, we would like to say something about the segment AF and AE, so note relationships among the following pair of similar triangle (the green area is just where the blue triangle BFC is overlapping triangle BEC):
From this figure the following relationships between similar triangles may be stated:
In looking at these triangles we have included a statement about every segment except BD and DC, and we have said nothing about segments common between pairs of similar triangle that may be used to relate these pairs to each other.† To attend to this, additional similar triangles are observed that seem to meet this need:
The following relationship may be stated:
Note that the segments PD and AP provide a link between the two relationships between similar triangles, so that all of these ratios are equal:
Now summarize the relationships we have discovered so far.
Inspect these three equations for the segments of interest.† The first term of equations (1) and (2) have four of the six segments in question (AF, BF, CE, and AE) so obviously we would like to keep these.† The only other two are CD and BD which appear in the first and fourth terms of equation (3).† The encouraging news is that if I decide to eliminate all but the first and fourth terms of equation (3), there are lengths AG which will link to equation (2) and AH which will link to equation (1).† Letís just hope that the length BC disappears.† So rewriting equations (1) through (3) based on the above discussion, we are left with:
(1)††††††† † or
Substitute (1) and (2) into (3):
As we had hoped, the length BC may be cancelled from both side of the equation.† The result becomes:
†††† or†††† , which, looking back at the original figure, is the result we are trying to prove.
A question arises as to whether this relationship holds for points P outside of the triangle.† Click hereto see an Animation of P moving along a path inside and outside of the triangle.† The demonstration shows that the relationship between the products, shown side by side, appears to hold regardless of the location of P.
A further investigation explores the relationship of the triangles ABC and DEF, where point P is within the boundary formed by ABC.† It is queried whether the ratio of the areas of ABC to DEF, with point P constrained in this way, is always greater than or equal to 4.† Where is equal to 4?† This is not proven here, but explored by demonstration.† Clear here to see the Demonstration.† As shown, the ratio of the areas is always greater than 4, except at a single point.† This point has the appearance of being the Centroid.† This is empirically confirmed by clicking the action button in the demonstration.