– Do you know what I got for my birthday from my wife?

– A coffee maker?

– Nope. Three line segments that measured 1, a, and b.

– How thoughtful of her. I guess you didn’t have them already.

– You are right. It made me extremely happy.

– How long did the happiness last?

– Till the next morning.

– What happened then?

– She asked me to make a line segment which measured a times b.

– I would have preferred the coffee maker.

Problem source: Pandalous.

Draw two intersecting lines L, M at random. Let them intersect at A. On line L construct B and C on opposite sides of A such that AB = a and AC = b. On line M construct D such that AD = 1. Construct the circle that goes through B, C and D. It intersects M at E where AE = ab.

To construct the circle: construct the perpendicular bisectors of the line segments BC and CD. These intersect at O, the centre of the circle. The circle’s radius is OB.

Cf:

Intersecting Chords theorem

@ Richard – thanks. lol – Can’t we just have the coffee instead?

– Adam

ha ha – kind of reminds me of René Magritte painting if a pipe with the inscription “This Is Not A Pipe” – Jill Copper

Who says the pyramids were built by extraterrestrials? Richard, I never would have thought of that…but I do follow your instructions of the theorem quite clearly.

I’m sure, also, that you have forgotten more geometry than I will ever remember.

Well done, sir.

It can be done quicklier using Thales’ Intercept Theorem:

A- – – -E- -C

¦

¦

D

¦

B

If AD is 1, AB is b and AE is a,

then, AC is ab!

—

Using the Intersecting Chord Theorem would be far better for this problem:

Given a segment of length 1 and another segment s, find a segment of length that is the square root of s.

(Which is the same as using the right triangle altitude formula* and Thales’ Circle Theorem)

*In a right triangle, the altitude with the hypotenuse as base divides the hypotenuse into two lengths p and q. If we denote the length of the altitude by h, we then have the relation h^2 = pq.

Wow, okay, you all went totally over my head! I guess I don’t remember my geometry from junior high. LOL! Either way, love coffee, not necessarily this machine. I def. drink too coffee to do one cup at a time.

Well that formula would certainly give me a headache if I could not have my coffee first. Maybe you will get lucky on next years birthday!

Why was this coffee machine worth mentioning in the first place? lol. I guess that’s just something you expected but, if she is the type of woman who does this instead…i just dont see the connection. Interesting story though.

lol. a lot of these replays are hilarious!!

Not a coffee machine? Well they tricked me