2D shapes and their properties

problem
Favourite

Baby Circle

Age

16 to 18

Challenge level

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?
problem
Favourite

Logosquares

Age

16 to 18

Challenge level

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.
problem
Favourite

Orthogonal Circle

Age

16 to 18

Challenge level

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.
problem

Area I'n It

Age

16 to 18

Challenge level

Triangle ABC has altitudes h1, h2 and h3. The radius of the inscribed circle is r, while the radii of the escribed circles are r1, r2 and r3 respectively. Prove: 1/r = 1/h1 + 1/h2 + 1/h3 = 1/r1 + 1/r2 + 1/r3 .
problem

Lawnmower

Age

14 to 16

Challenge level

A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?
problem

Fitting In

Age

14 to 16

Challenge level

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ
problem

Darts and Kites

Age

14 to 16

Challenge level

Explore the geometry of these dart and kite shapes!
problem

Retracircles

Age

16 to 18

Challenge level

Four circles all touch each other and a circumscribing circle. Find the ratios of the radii and prove that joining 3 centres gives a 3-4-5 triangle.
problem

Escriptions

Age

16 to 18

Challenge level

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
problem

Roaming Rhombus

Age

14 to 16

Challenge level

We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point D?