Categories
Intermediate

In and out

An obtuse triangle, of which α is the obtuse angle, with its incircle, two cevians and two tangents. Find a relation between α and β.

Categories
Beginner

Angle versus height

What is the ratio orange/green?

Categories
Intermediate

Wheel of fortune II

A triangle with its incircle and three cevians. The tangency points and incentre are shown. Find the relation between the angles α, β and γ.

Categories
Beginner

The Bedouin tent

A semicircle and a triangle of which one side is tangent to the semicircle. What is the angle α?

Categories
Intermediate

Six four

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

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

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?

Categories
Intermediate

Smugglers

The smugglers want to land half-way between the lighthouses. It is a dark night, so they keep their boat heading at an angle bisecting the angle to the two lights. Where on the shore should the coast guard wait to intercept them?