Problem of the Week #32: Monday August 14th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
This is a continuation of the Triangle Bisector problem I posted last week. Let’s say instead of “bisecting” the right triangle, we draw a line from the top vertex to the base with a length that is the average of the other leg and the hypotenuse, like this:
a). Does this line intersect the base on the left, on the right, or in the middle of the two lines we discussed in the last problem?
b). If the base of the triangle is one unit long, how far is this base intersection to the two in the last question?
c). Does this intersection become closer or further from each of the other two intersections as we increase the length of the base?
d). What is the maximum proportion of its distance from the midpoint of the base compared to the whole base for all right triangles in this position that have a height of 1 unit?
e). Does this intersection become closer or further from the other two as we decrease the measurement of the right angle (at the bottom left)?
f). What is the maximum proportion of its distance to the point where the angle bisector of the top vertex intersects the base compared to the whole base for all triangles (not necessarily right triangles) in this position and have both a base and a height of 1 unit?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.