A Big Doubt (Own Theorem)

A few years ago I discovered this property and it supposed to be my own Geometry Theorem. I want to know if is it real. Have you seen a Theorem like this?

As a corollary we can say:

If we let $a$ and $b$ be the foots of a right triangle with hypotenuse $c$ , we have

$\dfrac {c^{2}}{a^{2}} = \sum_{n=0}^\infty (\dfrac{b}{c})^{2n}$ And $\dfrac{c^{2}}{b^{2}}=\sum_{n=0}^\infty (\dfrac{a}{c})^{2n}$

#Geometry #HjalmarPhillippe #OwnTheorem?

Easy Math Editor

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Comments

You would need to explain the set up with some words, in order to explicitly convey what you intend.

Calvin Lin Staff - 5 years, 10 months ago

For the triangle in the above photo we know by euclidean theorem that $a^{2}=pc$ , and $h=\frac{ab}{c}$ . So it is true that - $a_{1} = \frac{c_{1}h}{b}$ - $a_{1}=\frac{\frac{b^{2}}{c}\frac{ab}{c}}{b}=a(\frac{b}{c})^{2}$ It's easy to see that $c_{1}=c(\frac{b}{c})^{2}$ by euclidean theorem. And the same with $b_{1}$ to get $b_{1}=b(\frac{b}{c})^{2}$ . If you take it with the next triangles at that side you'll find that $a_{n}=a(\frac{b}{c})^{2n}$ , $b_{n}=b(\frac{b}{c})^{2n}$ and $c_{n}=c(\frac{b}{c})^{2n}$ And with the triangles at the other side you'll find that $a_{k}=a(\frac{a}{c})^{2k}$ , $b_{k}=b(\frac{a}{c})^{2k}$ and $c_{k}=c(\frac{a}{c})^{2k}$

Hjalmar Orellana Soto - 5 years, 10 months ago

My calculations in Maple show me that it's true for triangles :

(a=4, b = 8.01, c = 8.95)

(a=3, b = 4, c = 5)

(a=4.9875077, b=12.429690, c=13.392999)

John Doe - 5 years, 10 months ago

The corollary or the theorem?

Hjalmar Orellana Soto - 5 years, 10 months ago

I think it's theorem, I have not seen it before.

John Doe - 5 years, 10 months ago

I've seen to problem of this sort. The closed form of $a_{k}$ wasn't given, but we had to sum a very similar series.

A Former Brilliant Member - 5 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$