MathDB

Let

ABCD

be a quadrilateral with all the internal angles

< \pi

. Squares are drawn on each side as shown in the picture below. Let

\Delta_1 , \Delta_2 , \Delta_3 , \Delta_4

denote the areas of the shaded triangles as shown. Prove that

\Delta_1 - \Delta_2 + \Delta_3 - \Delta_4 = 0.

[asy] //made from sweat and hardwork by SatisfiedMagma import olympiad; import geometry;
size(250);
pair A = (-3,0); pair B = (0,2); pair C = (5.88,0.44); pair D = (0.96, -1.86);
pair H = B + rotate(90)*(C-B); pair G = C + rotate(270)*(B-C);
pair J = C + rotate(90)*(D-C); pair I = D + rotate(270)*(C-D);
pair L = D + rotate(90)*(A-D); pair K = A + rotate(270)*(D-A);
pair F = A + rotate(90)*(B-A); pair E = B + rotate(270)*(A-B);
draw(B--H--G--C--B, blue); draw(C--J--I--D--C, red); draw(B--E--F--A--B, orange); draw(D--L--K--A--D, magenta); draw(L--I, fuchsia); draw(J--G, fuchsia); draw(E--H, fuchsia); draw(F--K, fuchsia);
pen lightFuchsia = deepgreen + 0.5*white;
fill(D--L--I--cycle, lightFuchsia); fill(A--K--F--cycle, lightFuchsia); fill(E--B--H--cycle, lightFuchsia); fill(C--J--G--cycle, lightFuchsia);
label("

\triangle_2

", (E+B+H)/3); label("

\triangle_4

", (D+L+I)/3); label("

\triangle_3

", (C+G+J)/3); label("

\triangle_1

", (A+F+K)/3);
dot("

A

", A, S); dot("

B

", B, S); dot("

C

", C, S); dot("

D

", D, N); dot("

H

", H, dir(H)); dot("

G

", G, dir(G)); dot("

J

", J, dir(J)); dot("

I

", I, dir(I)); dot("

L

", L, dir(L)); dot("

K

", K, dir(K)); dot("

F

", F, dir(F)); dot("

E

", E, dir(E)); [/asy]

P3

Problems(1)