examday

/TMUA

A triangle is inscribed in a circle so that its vertices lie on the circumference. How many of the following statements are individually sufficient to imply that it is an acute-angled triangle (i.e. all of its angles are strictly between $0$ and $\dfrac\pi2$ radians)?

• 1. None of its sides is a diameter of the circle
• 2. Every diameter of the circle meets the triangle
• 3. If the triangle is reflected in its longest edge, the reflected triangle meets the circle
• 4. If the triangle is reflected in its longest edge, the reflected triangle meets the circle at least three times
• 5. Its lengths and angles satisfy the cosine rule
• 6. The centroid of the triangle is the centre of the circle.