Categories
Intermediate

Wedging the circle

Four similar triangles share one circle as incircle or excircle. Show that orange dotted lines must be concurrent.

Categories
Intermediate

Interlocked triangles

Do the blue, green, and orange triangles have the same shape?

Categories
Intermediate

Six four

A regular hexagon and an inscribed tilted rectangle. What is the angle α?

Categories
Intermediate

Ball of yarn

A cyclic hexagon has concurrent cross diagonals. What is the ratio of the product of the orange sides compared to the product of the purple sides. (ace:bdf).

Categories
Intermediate

Sunrise over green mountain

An acute triangle mountain ABC has altitudes BE and CF. The dotted tangent lines to the sunny circumcircle (AEF) at E and F intersect at a point M. Show that M is on the mountain’s base BC.

Categories
Intermediate

The silly walk

A circle and several triangles. Prove that the green triangle is isosceles.

Categories
Intermediate

The sun hat

Two triangles share a circumcircle and vertex, with one edge of the orange triangle containing the feet of two of the altitudes of the blue triangle. Show that the orange triangle is isosceles.

Categories
Intermediate

Uneasy the head

The green zigzag crown segments would extend through either B or C. Show that the arcs along the top are equally spaced.

Categories
Intermediate

Giant’s shoulder II

Three equilateral triangles share three vertices. What is blue : red?

Categories
Intermediate

Falling in

Start with an acute triangle and form a new triangle from the points of tangency of its inscribed circle. Continue this process to make make the triangle with blue vertices. What is the maximum possible angle at a blue vertex?