1. In Triangle ABC, the circumcenter and orthocenter are collinear with vertex A. Which of the following statements must be true?

(1) Triangle ABC must be an isosceles triangle.

(2) Triangle ABC must be an equilateral triangle.

(3) Triangle ABC must be a right triangle.

(4) Triangle ABC must be an isosceles right triangle.

Enter your answer as a comma-separated list. If there is no correct option, write "none".

2.

Let H be the orthocenter of the equilateral triangle ABC. We know the distance between the orthocenters of Triangle AHC and Triangle BHC is 12. What is the distance between the circumcenters of Triangle AHC and Triangle BHC?

4.

The diagonals of a trapezoid are perpendicular and have lengths 3 and 4. Find the length of the median of the trapezoid.

Guest Dec 13, 2018

#1

#2**0 **

In some cases, (2) also works because an equilateral triangle is a special type of isosceles triangle.

PartialMathematician
Dec 13, 2018