The rectangles are 3 by 4. Which of the three bounding octagons have similar shape?

# Author: Marshall W. Buck

## Shooting the moon

Show that the blue dots are concyclic and also the orange dots are concyclic.

## Fan of cards

N coloured rectangular cards are arranged with a common vertex and common angle α between cards. If a single card has perimeter P, what is the perimeter of the coloured region, in terms of N, P, and α?

Extra condition: N and α are such that at least two edges of the bottom card are visible.

## The telescope

The odd regular polygons are stacked together, in increasing order, zigzagging to form a telescope that extends forever! What is the limiting angle (with respect to the horizontal) at which the telescope points?

## The weight lifter

Express the x coordinate of E in terms of x coordinates of A, B, F, and G. The two black lines are supposed to be parallel to the x-axis.

## Encircling an anvil

Two equilateral triangles, one upside down, have boundaries meeting at 4 points, 3 of which are marked in red. Show that the circle through the red points also passes through the centre point of the blue triangle.

## Bullet train

What is the ratio of orange and green areas? (The orange triangle is 45-45-90)

## Roly poly doll

Two circles with diameters (green and orange) from one of the intersection points. Show that the other intersection point of the circles is on the (purple dotted) line connecting the other ends of the diameters.

## Flying purple people eater

What is the area of the purple people eater, in terms of the lengths AD and BC?

## Mountain roads

A circular mountain peak has coordinates (0,0,3) and its base has towns C=(-4,0,0), E=( 4,0,0) on opposite sides. Town D=(0,4,0) is halfway between them, around the base. A road goes partially up and down the mountain connecting C and E. Another road connects C and D. Both roads are shortest possible. What is the ratio of the road lengths?