The blue triangle is similar to the green one and has the same orientation. The midpoints between corresponding vertices are connected to form a red triangle. Prove that it is also similar.

# Category: Advanced

## The hawfinch II

A circle with two tangents. What’s red : blue in terms of α?

## The weather balloon

A circle with three red tangents. The tangency points are marked red. Prove that the yellow triangle is isosceles.

## Count to five

A triangle with a cevian. What’s α?

## The orange pennant

A regular pentagon with two coloured triangles. What’s the proportion orange : blue?

## On a tangent

Two tangent circles and their three common tangents. Two secants meet on the internal tangent. Prove that the three red points are collinear.

## The new normal

Two equilateral triangles are placed as shown. Their centres are connected as well as their intersection points. What’s the angle α?

## The Moroni minimum

An arbitrary point on the parabola y=x^{2}. Shown are the tangent in that point and a line segment perpendicular to it. What is the minimal height h?

## The common ground

Two squares share a vertex. Two other vertices are 4 units apart. A third green square has the centres as opposite vertices. What is its area?

## Çetins circle

A square, a semicircle and six line segments. What’s the angle α?