Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Vecten

Stage: 5 Challenge Level:

Aleksander Twarowski from III LO Gdynia sent us his essay on this topic.

 Aurel from King Edward VI School, Southampton, has proved Part (1) that the four triangles $r$, $s$, $t$ and $D$ have the same area. Can you follow some of the suggestions given for the other parts and discover anything else? Here is Aurel's solution: Let the sides opposite angles $A$, $B$ and $C$ be $a$, $b$ and $c$ respectively. The length of each side of all three squares are therefore known.
Consider triangle $D$. Using the well known formula, the area of triangle $D$ is given by: $$\Delta = {1\over 2}ab\sin C = {1\over 2}bc\sin A={1\over 2}ca\sin B$$ where all three versions of the formula are clearly equivalent because the labelling of the triangle was arbitrary in the first place.

Consider triangle $s$. The angle $B^*$ at $B$ in this triangle can be found by considering that the angles around a point add up to $360^o$, so $\angle B^*=360 - 90 -90- B = (180 - B)$.

Therefore, the area of triangle $s$ is ${1\over 2}ac\sin (180-B)$.

By exactly the same reasoning the area of triangle $r$ is ${1\over 2}bc\sin (180-A)$ and the area of triangle $t$ is ${1\over 2}ab\sin (180-C)$.

In general we know that $\sin (180 - \theta) = \sin \theta$ so:

the area of triangle $s$ is ${1\over 2}ac\sin (180-B)= {1\over 2}ac\sin B$
the area of triangle $r$ is ${1\over 2}bc\sin (180-A)= {1\over 2}bc\sin A$
the area of triangle $t$ is ${1\over 2}ab\sin (180-C)= {1\over 2}ab\sin C$

So triangles $r$, $s$ and $t$ all have the same area as triangle $D$ and so we have shown that all four triangles have the same area.

The problem has also been discussed on AskNRICH .