MathDB

The opposite sides of the reentrant hexagon

AFBDCE

intersect at the points

K,L,M

(as shown in the figure). It is given that

AL = AM = a, BM = BK = b

CK = CL = c, LD = DM = d, ME = EK = e, FK = FL = f

. http://imgur.com/LUFUh.png

(a)

Given length

a

and the three angles

\alpha, \beta

and

\gamma

at the vertices

A, B,

and

C,

respectively, satisfying the condition

\alpha+\beta+\gamma<180^{\circ}

, show that all the angles and sides of the hexagon are thereby uniquely determined.

(b)

Prove that

\frac{1}{a}+\frac{1}{c}=\frac{1}{b}+\frac{1}{d}

Easier version of

(b)

. Prove that

(a + f)(b + d)(c + e)= (a + e)(b + f)(c + d)

3