MathDB

In the diagram,

ABDF

is a trapezoid with

AF

parallel to

BD

and

AB

perpendicular to

BD.

The circle with center

B

and radius

AB

meets

BD

C

and is tangent to

DF

E.

Suppose that

x

is equal to the area of the region inside quadrilateral

ABEF

but outside the circle, that y is equal to the area of the region inside

\triangle EBD

but outside the circle, and that

\alpha = \angle EBC.

Prove that there is exactly one measure

\alpha,

with

0^\circ \leq \alpha \leq 90^\circ,

for which

x = y

and that this value of

\frac 12 < \sin \alpha < \frac{1}{\sqrt 2}.

[asy] import graph; size(150); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen fftttt = rgb(1,0.2,0.2); draw(circle((6.04,2.8),1.78),qqttff); draw((6.02,4.58)--(6.04,2.8),fftttt); draw((6.02,4.58)--(6.98,4.56),fftttt); draw((6.04,2.8)--(8.13,2.88),fftttt); draw((6.98,4.56)--(8.13,2.88),fftttt); dot((6.04,2.8),ds); label("

B

", (5.74,2.46), NE*lsf); dot((6.02,4.58),ds); label("

A

", (5.88,4.7), NE*lsf); dot((6.98,4.56),ds); label("

F

", (7.06,4.6), NE*lsf); dot((7.39,3.96),ds); label("

E

", (7.6,3.88), NE*lsf); dot((8.13,2.88),ds); label("

D

", (8.34,2.56), NE*lsf); dot((7.82,2.86),ds); label("

C

", (7.5,2.46), NE*lsf); clip((-4.3,-10.94)--(-4.3,6.3)--(16.18,6.3)--(16.18,-10.94)--cycle); [/asy]

Problem Statement