Final Project

by Jongsuk Keum

A. Consider any triangle ABC. Select a point P inside the triangle and draw lines AP, BP, and CP extended to their intersections with the opposite sides in points D , E , and F respectively.

Explore (AF) (BD) (EC) and (FB) (DC) (EA) for various triangles and various locations of P.
Conjecture ?
The ratio
(AF) (BD) (EC) / (FB) (DC) (EA) = 1
Let angle AFP = x, angle BDP = y, angle CEP = z , angle DPC = p, angle BPF = q , angle APE = r.
Let's consider the area of small triangles.
  AFP = 1/2(AF)(FP)sinx, AFP = 1/2(AP)(FP)sinp---------------------1
  BFP = 1/2(BF)(FP)sinx, BFP = 1/2(BP)(FP)sinq---------------------2
BPD = 1/2(BD)(DP)siny, BPD = 1/2(BP)(DP)sinr-------------------3
CPD = 1/2(CD)(DP)siny, BPD = 1/2(CP)(DP)sinp------------------4
 PCE = 1/2(PE)(EC)sinz, PCE = 1/2(PE)(PC)sinq--------------------5
  APE = 1/2(PE)(EA)sinz, PCE = 1/2(PE)(AP)sinr---------------------6
By dividing 1,3,5 by 2,4,6 respectively, we get
AF/BF = (AP/BP)sinp/sinq
BD/BC = (BP/CP)sinr/sinp
CE/AE = (CP/AP)sinq/sinr
Therefore, by multipling each side, we get
AF BD CE / BF DC AE = 1   Q.E.D.

B. Can the result be generalized (using lines rather than segments to construct ABC) so that point P can be outside the triangle? Show a working GSP sketch.

Click here to see a GSP sketch.

C. Show that when P is inside triangle ABC, the ratio of the areas of triangle DEF is always grater than or equal to 4. When is it equal to 4?

Generally, the ratio of the areas of triangle DEF is always grater than or equal to 4.
When P is a centroid of triangle ABC, it is equal to 4.
Triangle ABC is similar to triangle DEF, and AB: DE = BC: EF = CA : FD = 2 : 1.
So the ratio of areas is equal to 4.  Q.E.D.

Click here to see a GSP file.

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