Learn – Geometry – Miscellaneous

Problem set 1

In the following figure, $O$ is the center of the circle, and $AB$ is a diameter of the circle. Prove that $\angle ACB = 90^\circ$ .

Problem set 2

In this problem set, two concentric circles are shown in the figures. $O$ is the center of the circles.

In the following figure, prove that $\frac{EB}{OE} = \frac{DC}{OD}$
In the following figure, prove that $\frac{DB}{OB} = \frac{CA}{OA}$
In the figure above, prove that $\frac{DF}{OD} = \frac{CE}{OC}$ .
In the figure above, prove that $\frac{DF}{OB} = \frac{CE}{OA}$ .
In the figure above, prove that $\frac{BD+DF}{OB} = \frac{AC+CE}{OA}$ .
In the figure above, prove that $\frac{BD+DF+FH+HJ+JL}{OB} = \frac{AC+CE+EG+GI+IK}{OA}$ .
Do you see how $\frac{\mbox{arc length }BL}{\mbox{radius }OB} \approx \frac{\mbox{arc length }AK}{\mbox{radius }OA}$ ?
Do you see how $\frac{\mbox{circumference of big circle}}{\mbox{radius of big circle}} \approx \frac{\mbox{circumference of small circle}}{\mbox{radius of small circle}}$ ?
Do you see how $\frac{\mbox{circumference of big circle}}{\mbox{diameter of big circle}} \approx \frac{\mbox{circumference of small circle}}{\mbox{diameter of small circle}}$ ?