## Saturday, March 21, 2020

### Dynamic Geometry Problem 1464: Quadrilateral, Interior Point, Midpoint of Sides, Equal Sum of Areas

Interactive step-by-step animation using GeoGebra. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below. 1. From the midpoint theorem,
EF//AC//GH and FG//BD//EH so EFGH is a parallelogram and by Problem 1463,

S(PEF) + S(PGH) = S(PFG) + S(PEH) ....(1)

Now S(ABCD) = S(ABC) + S(ACD) = 4.S(BEF) + 4.S(DGH) ...(2)

Similary S(ABCD) = S(ABD) + S(BCD) = 4.S(AEH) + 4.S(CFG) ....(3)

From (2) & (3), S(BEF) + S(DGH) = S(CFG) + S(AEH) ...(4)

S2 + S4 = S3 + S1

Sumith Peiris
Moratuwa
Sri Lanka

2. APE(=PEB+)+APH(=PHD+)+CPG(=GPD+)+CPF(=FPB+)
=> S yellow(=S green)

1. Good solution, much simpler than mine

2. Thanks you are big favorite of me for your solutions

3. Thank u. Same here!

3. P1465
AE=AH, BE=BF, CG=FC, DG=DH => AE+BE+CG+DG=AH+BF+FC+DH =>
(AE+BE+CG+DG).R =(AH+BF+FC+DH).R
S yellow = S green