As always, if you want to run off and do this proof on your own, look at the statement below and then come back:
Prove that the sum of the measures of the perpendiculars from any point on a side of a rectangle to the diagonals is constant.
The animation (code here) on the right is a demonstration of what you're looking to prove (from here, once again). The red and blue segments show "all" the perpendiculars drawn to the diagonals of this rectangle as a point moves around the perimeter. The "scale" below the diagram represents the changing lengths of these perpendiculars. As you can see, the sum of these lengths is constant.
A "Staring Out the Window" Kind of Proof
I'll start in the middle. No matter where I draw a point on a rectangle, I can draw two perpendiculars—one to each diagonal—so long as I'm cool with a perpendicular that has a length of 0 at each corner.
And drawing that point also divides the side on which it is drawn into two segments, which I've labeled as \(\small\mathtt{w - b}\) and \(\small\mathtt{b}\) here, using \(\small\mathtt{w}\) for the width of the rectangle.
Further, I always create two similar triangles. The two right triangles with sides P1 and P2 have another pair of corresponding congruent angles, \(\small\mathtt{a}\), and are thus similar by AA Similarity.
Since this is the case, the ratios of the corresponding sides are equal, and so \(\mathtt{\frac{b}{w - b} = \frac{P1}{P2}}\). And quickly following from this, we have: \[\small\mathtt{(w - b)P1 = bP2}\] \[\small\mathtt{wP1 - bP1 = bP2}\] \[\small\mathtt{wP1 = bP1 + bP2}\] \[\small\mathtt{\color{purple}{w \cdot \frac{P1}{b} = P1 + P2}}\]
This last equation can be written as \(\small\mathtt{w \cdot sin(a) = P1 + P2}\).
Turning our attention to the height, \(\small\mathtt{h}\), of the rectangle, we can do almost exactly the same as we did with the width (and we'll reuse variables to do it). Here we've got \(\mathtt{\frac{b}{h - b} = \frac{P1}{P2}}\). This gives us, in turn, \[\small\mathtt{(h - b)P1 = bP2}\] \[\small\mathtt{hP1 - bP1 = bP2}\] \[\small\mathtt{hP1 = bP1 + bP2}\] \[\small\mathtt{\color{purple}{h \cdot \frac{P1}{b} = P1 + P2}}\]
Here, \(\mathtt{\frac{P1}{b}}\) is equal to sin(z), which is cos(a), since \(\small\mathtt{z}\) = 90\(\small\mathtt{^\circ - a}\). So, we can rewrite that last equation as \(\small\mathtt{h \cdot cos(a) = P1 + P2}\).
Now we've got \(\small\mathtt{w \cdot sin(a) = P1 + P2}\) and \(\small\mathtt{h \cdot cos(a) = P1 + P2}\). And since \(\small\mathtt{w \cdot sin(a) = h \cdot cos(a)}\) (can you see how?), we see that \(\small\mathtt{P1 + P2}\) is constant all around the rectangle. Investigating each of the remaining two sides will yield one of those two equations.
What's Missing?
There are some missing pieces, but that's okay. One of the reasons I started writing this notebook—and populating it with mostly proofs—is that I wanted to write (and write up) proofs in a way I wanted to read them: as narratives, as highly persuasive mathematical essays, in a sense. And good essays don't fill in every blank for you. No information does.
My sense is that students would have a better time with them as well. If we asked them (occasionally, not wall-to-wall pedal-to-the-metal all-grades-everywhere) to "make a case" for one geometric relationship or another, we could bridge this divide between la la la la math on the one hand and ERMAHGERD PROOF! on the other. And thinking about the goal for "case making" in this way can help us focus on good reasons for getting kids to explain themselves in earlier grades.