The drawing is very simple:

We have a triangle ABC with C=90. Complete it to a rectangle ACBD.

Inscribe an incircle into ABC. Draw parallels through incenter

to the rectangle sides. You get a smaller rectangle that shares

vertex D with ACBD, (it overlaps ABC over a small triangle),

the small rectangle has area that equals to ABC,

the sides of the small rectangle are the segments of hypotenuse made

by tangency point of incircle (they also equal to radii of ex-circles).

The core of the proof is in the expression (p-a)*(p-b)=a*b/2

where p = (a+b+c)/2 In doing multiplication for the equality

to hold the c^2 = a^2 + b^2 must be true.

This proof is very close to #33 and #42.

I hope this time you got the configuration correctly

(probably you constructed point D as reflection of C in AB,

instead of antipode, or reflection in circumcenter)

so there should be no need in the file, but anyway

how do I send a drawing in a Sketchpad file ?

Salute,

Maj. Pestich

You can attache a file to your message. Just below the edit control into which you type your messages there us an Attachment box with the words "Click here to choose your file" beneath it. The way to do that is to "Click here" and choose the file.

If possible make if a graphics file, not Sketchpad's.

Thank you.
Alex

the proof # 42, so just forget about it.

It is hard to come up with something different

from what's in the list already, if the proof is

from 90 deg. angle to c*c = a*a + b*b.

It is more challenging to prove it in the direction

from cc = aa + bb to the 90 deg. angle.

For example, in a circle draw a diameter AB and a

chord CD parallel to this diameter. Take a fixed point X

on diameter (not the center) and connect it to

the ends of the chord ( line segments XC, XD).

Then XC^2 + XD^2 is invariant (obviously equal

to AX^2 + BX^2. Can make a proof of P.theorem

from this configuration, but need other than P. theorem

tools to prove orthogonality.

Also, in triangle ABC with altitudes AD from A and BE

from B we have AB^2 = AC * AE + BC * BD.

Seems to be a good raw material for P. proof.

I still do not know how to prove P.T. from

parabola (to show that x-coordimates make 90 deg

triangle when y-coordinates are Yc = Ya + Yb).

A whole lot of identities can be turned into

P. theorem proof. It's like distilling alcohol,

one can do it practically from anything,

including kitchen tabouret, some even prefer

'tabouretovka' taste to 'plain vanila' wheat grains.

It was much easier for Euclid, he had the proper

sequence of the flow from one theorem to the other,

he knew exactly what follows from what. We were

taught from different

textbooks, with different flows, that is why there are

some proofs that use power of a point and inversion

to prove P. theorem, although most inversion stuff

has P. theorem. as it's foundation.