Mean Proportional in Right Triangles. Geometry video.

Leg rule proof. Right triangle properties.
#geometry #righttriangles #proportions

[AnonimityAssured]

I had never before heard of the concept of projection applied to such a context. I learn so much from you.

Reflecting on this problem while lying in bed last night, it occurred to me that it might form the basis of a simple proof of the Pythagorean Theorem.

To adopt the standard labelling, we can replace x with a, and label the vertical leg b. We can also replace 2 with p and 6 with q, and say that p + q = c.

p + q = c;
(p + q)² = c²;
p² + 2pq + q² = c².

By similar triangles:

a / (p + q) = p / a;
a² = p(p + q);
= p² + pq.

Similarly:

b / (p + q) = q / b;
b² = q(p + q);
= q² + pq.

Combining these results:

a² + b² = p² + 2pq + q² = c²;
a² + b² = (p + q)² = c².

So, a² + b² = c², which is the Pythagorean Theorem. (Q.E.D.)

I could have dotted more i's and crossed more t's, but I think the procedure is clear.