# 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
Proof.
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.