Pumping triangles

Question

On the inside of the triangle (0,0),(1,0),(0,1), define the "pretty useless map" or pump by the following prescription:

Given a point A find its projections x0,y0 to the x and y axes and also x1,y1 to the same axes shifted by one unit. Connect x1 to y0 and y1 to x0. The intersection of these two straight lines shall be pump(A)
Question:

Given a triangle ABC show that the area of triangle pump(A)pump(B)pump(C) can be written as f(X) for some fixed scalar function f where X is the area of ABC.

Most elegant solution wins.
Hint 1:

Hint 2 (Warning: this really is a spoiler):

Source: Me.

greenturtle3141 · Accepted Answer

The alternative solution:

Then:

Gareth McCaughan · Answer

Here's a short but rather unenlightening algebraic proof. (I thought I had a nice simple more-geometric proof but it has a hole, so while I think about filling the hole I'll write this one out so there's a solution.)

Answered by Gareth McCaughan on August 8, 2021

Pumping triangles

2 Answers

Add your own answers!

Ask a Question