AN INTERESTING RATIO RESULT FOR TRIANGLES

by

SriRanga S. Dattatreya
and
Ravi E. Dattatreya

SUMMARY

We show a construction and a general ratio result for triangles that can be used to segment a line into various parts. The construction has as a special case the GLaD construction. (Mathematics Teacher, January 1997).

CONSTRUCTION

For any triangle ABC, select an arbitrary point M. Join AM, BM and CM, and extend them to cut BC, AC and AB at D, E and F, respectively. Join DE. Let the point of intersection of CF with DE be N. Join AN and extend it to meet CB at G.

THEOREM

GC/DG = CD/BD + 1

PROOF

Applying Ceva's theorem to triangle ABC and the point M, we obtain:

AF/FB . BD/CD . CE/EA = 1

Also, from Jim Wilson's result, we have:

AM/MD = AE/CE . (CD/BD + 1)

Now, apply Ceva's theorem to triangle ADC and the point N. We have:

AM/MD . DG/GC . CE/EA = 1

Substituting for AM/MD,

(AE/CE . (CD/BD + 1)) . DG/GC . CE/EA = 1

Simplifying,

(CD/BD + 1) . DG/GC = 1

From this:

GC/DG = CD/BD + 1

Q.E.D.

APPLICATION

If D is the midpoint of BC, that is, AD is a median, and M is any point on the median, then we have:

CD/BD = 1

and, therefore,

GC/DG = 2

In other words,

CG = 1/3 . BC

If we repeat the construction with N as the point, we can obtain the point H such that

BH = 1/4 . BC

and so on. Thus, we can segment the line BC into integral divisions without drawing any parallel lines. Just the location of the midpoint D of BC is required. (Picture)

If we extend the point B to infinity along CB, then the line ME will become parallel to BC. Also, if we extend the point A to infinity along BA, DM and GN, which are segments of DA and GA, will become parallel. Then, our construction begins to resemble a GLaD-like construction applied to the quadrilateral DMEC which has actually become a parallelogram (see also Jim Wilson's comments on the GLaD construction.)

FUTURE WORK

In exploring the literally hundreds of various distances, ratios and areas that result in repeating this construction with the Geometer's Sketchpad, we found very interesting relationships. Perhaps some of these are worth exploring.

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APPENDIX A: ILLUSTRATION