Problem:
The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are , , and
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Consider the altitude from to . This line has equation .
Consider the altitude from to . The slope of this line is the negative reciprocal of the slope of .
The slope of , which means the slope of is .
Thus the equation of the altitude through is:
and plugging in gives , so =
The problems on this page are the property of the MAA's American Mathematics Competitions