Thanks to Willem for the suggestion!

Point E is exterior to square ABCD with EA = 6, EB = 3, and EC = 5. What is the area of square ABCD?

As usual, watch the video for a solution.

Or keep reading.

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Answer To Area Of Square From An External Point

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Suppose the square has a side length equal to x.

Suppose the horizontal distance from B to E is a and the vertical distance is b. We can then construct 3 triangles from point E whose hypotenuses are the three lengths 6, 3, and 5.

By the Gougu Theorem, we have the following three equations:

(I)
a² + b² = 3² = 9

(II)
(a + x)² + b² = 5² = 25

(III)
a² + (b + x)² = 6² = 36

In equation (II) we will solve for a. We will use a² + b² = 4 from (I) in the second step.

(a + x)² + b² = 25
x² + 2ax + a² + b² = 25
x² + 2ax + 9 = 25
a = (16 – x²)/(2x)

We will similarly solve for b in (III):

a² + (b + x)² = 36
a² + b² + 2bx + x² = 36
9 + 2bx + x² = 36
b = (27 – x²)/(2x)

We can now substitute for a and b in (I):

a² + b² = 9
((16 – x²)/(2x))² + ((27 – x²)/(2x))² = 9
(16 – x²)² + (27 – x²)² = 36x²
256 – 32x² + (x²)² + 729 – 54x² + (x²)² = 36x²
2(x²)² – 122x² + 985 = 0

The area of the square is x². The last equation is quadratic in x², so we can solve for its value using Brahmagupta’s quadratic formula, giving:

x² = [-(-122) ± √((-122)² – 4(2)(985))]/(2(2))
x² = 61/2 ± (√1751)/2

Usually in geometry problems we want the positive radical to get a positive length. But in this problem we have two positive values:

x² = 61/2 + (√1751)/2 ≈ 51.4
x² = 61/2 – (√1751)/2 ≈ 9.6

The key in this problem is we want E to be exterior to the square, so we want a and b to be non-negative values. Recall that:

a = (16 – x²)/(2x)

For a to be non-negative, we need x² ≤ 16.

Thus we must exclude the area ≈ 51.4 case. The correct answer is thus:

Area = x² = 61/2 – (√1751)/2 ≈ 9.6 units squared.

This problem was on Reddit AskMath and the picture has a credit to Logic Booster. The poster thought there was insufficient information. But people replied that it was possible to solve. Can you figure it out?

As usual, watch the video for a solution.

Right Triangle Area Puzzle

Or keep reading.

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Answer To Right Triangle Area Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Method 1: Algebra

Let the legs be a and b and the hypotenuse c = 64 cm. Because the perimeter is 150 cm we have:

a + b + c = 150
a + b + 64 = 150
a + b = 86
(a + b)² = 86²
a² + b² + 2ab = 86²

But since we have a right triangle we have:

a² + b² = c² = 64²

Substituting we have:

64² + 2ab = 86²
2ab = 86² – 64²
ab/2 = (86² – 64²)/4
ab/2 = (86 + 64)(86 – 64)/4
ab/2 = 825

So the area is 825 cm², and we didn’t have to solve for the sides of the triangle!

Method 2: Inradius

An alternate method is to construct the incircle with a radius r. In any triangle the area is:

rp/2

Furthermore, in a right triangle the hypotenuse is equal to:

c = a – r + b – r

So we can add c to both sides and get:

2c = a + b + c – 2r
2c = p – 2r
r = (p – 2c)/2

So now let’s solve the original problem.

We can solve for the inradius using the given information:

r = (p – 2c)/2
r = (150 – 2(64))/2
r = 11

Then we have the area of the triangle is:

rp/2
= 11(150)/2
= 825

So we again get the answer of 825 cm².

Extra credit

Solve for the lengths of the legs.

Reference

Reddit AskMath
https://www.reddit.com/r/askmath/comments/1tu6zwa/found_on_facebook_not_enough_info/

a² + b² = c²

But who actually discovered this formula, and how did they figure it out?

The following video presents the earliest proofs from China, India, and Greece (Euclid), as well as a few tangents to modern proofs including President Garfield’s proof and Albert Einstein’s proof.

Who Really Discovered The Pythagorean Theorem?

I spent a long time on the visuals and animations for the video, so I have not prepared a text version with static graphics as I normally do. I hope you can watch the video when you have time. Here are some time stamps that you might find useful:

0:00 Intro
1:50 China
5:00 India
7:35 Garfield
8:45 Pythagoras
9:24 Greece
15:18 Einstein

References

Community post
https://www.youtube.com/post/UgkxGSwX5dtLg6zWkm-rNohQdcgDduhp5Gi2

Babylon, Egypt
https://en.wikipedia.org/wiki/File:Plimpton_322.jpg
https://en.wikipedia.org/wiki/File:Papyrus_Berlin_6619.jpg
https://commons.wikimedia.org/wiki/File:All_Gizah_Pyramids.jpg
https://commons.wikimedia.org/wiki/File:Corda-a-12-nodi.svg
Ricardo Liberato, CC BY-SA 2.0 , via Wikimedia Commons

China (Gougu)
https://en.wikipedia.org/wiki/Xuan_tu
https://en.wikipedia.org/wiki/Zhoubi_Suanjing
https://en.wikipedia.org/wiki/File:Duke_of_Zhou.png
https://www.rediff.com/news/special/did-india-discover-pythogoras-theorem-a-top-mathematician-answers/20150109.htm

India (Baudhayan)
https://en.wikipedia.org/wiki/Baudhayana#/media/File:Bas-relief_of_Baudhayana_(cropped).jpg
https://archive.org/details/baudhyanasrautas01bauduoft/page/n5/mode/2up
https://en.wikipedia.org/wiki/Yajna
Arayilpdas at Malayalam Wikipedia CC BY-SA 3.0 via Wikimedia Commons
https://w.wiki/S$M5
https://x.com/ncert/status/2006602652638130291
https://ncert.nic.in/textbook/pdf/hegp202.pdf
https://bhaskarkamble.substack.com/p/the-sulbasutras-and-the-pythagoras

Greece (Pythagoras)
https://commons.wikimedia.org/wiki/File:Pythagoras_knapp.jpg
https://archive.org/details/cu31924008704219/page/n165/mode/2up

Greece (Euclid)
https://mathcs.holycross.edu/~little/Mont201516/EuclidPythagoras1.html
https://commons.wikimedia.org/wiki/File:Tysoe_Windmill_-_geograph.org.uk_-_7335000.jpg
Tysoe Windmill by Derek Harper, CC BY-SA 2.0 , via Wikimedia Common
https://www.cut-the-knot.org/pythagoras/euclid.shtml
https://webspace.ship.edu/mrcohe/inside-out/vu1/d_joyce/elements/bookVI/propVI31.html
http://aleph0.clarku.edu/~djoyce/elements/bookVI/propVI31.html
https://en.wikipedia.org/wiki/File:Visual_proof_of_the_Pythagorean_theorem_by_area-preserving_shearing.gif
https://www.desmos.com/calculator/zlf3dweki4
https://www.youtube.com/watch?v=eWNBw5GpqeY&t=611s

Einstein
https://archive.org/details/fractalschaospow0000unse/page/2/mode/2up
https://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtml
https://terrytao.wordpress.com/2007/09/14/pythagoras-theorem/
https://www.newyorker.com/tech/annals-of-technology/einsteins-first-proof-pythagorean-theorem

Impossible proof
https://www.tandfonline.com/doi/epdf/10.1080/00029890.2024.2370240?needAccess=true
https://maa.org/news/groundbreaking-pythagorean-research-featured-in-the-american-mathematical-monthly/
https://www.youtube.com/watch?v=juFdo2bijic