# Monge's Theorem and the Euler Line **URL:** \ **Category:** Factual Questions\ **Created:** [August 5, 2025, 1:32am UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707 "2025-08-05T01:32:20Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [August 5, 2025, 1:32am UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/1 "2025-08-05T01:32:21Z")
Monge’s Theorem states that, when you graph the external tangent line linking three circles in a plane…none of which are inside each other…the intersection points of the tangents lie in a straight line. [Monge’s theorem - Wikipedia](https://en.wikipedia.org/wiki/Monge%27s_theorem) [Monge’s Theorem - YouTube](https://www.youtube.com/shorts/zxOK2vKVfQM) When you take a triangle’s circumcenter, orthocenter, and centroid, those points also lie in a line. [Euler line - Wikipedia](https://en.wikipedia.org/wiki/Euler_line) [Euler Line #maths #mathematics - YouTube](https://www.youtube.com/shorts/3xY7MG6mX9M) Lines based on the geometry of three. Are the two lines mathematically related? ---
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [August 7, 2025, 6:01pm UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/2 "2025-08-07T18:01:59Z")
Here’s a bit more obscure geometric shennanigans regarding tangents and the center of ellipses lying along a line. (relevant part starts around 7:40) [Extraordinary Conics: The Most Difficult Math Problem I Ever Solved](https://www.youtube.com/watch?v=X83vac2uTUs) ---
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [August 11, 2025, 1:24pm UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/3 "2025-08-11T13:24:36Z")
Did I stump the great minds of the Straight Dope Message Board? Chronos? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [August 11, 2025, 2:04pm UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/4 "2025-08-11T14:04:35Z")
“Are these things mathematically related?” is a difficult question to answer, because in some sense, all things in mathematics are related. But that’s probably not what the questioner means. You could also parse it as “Do the proofs of these two theorems have a similar structure?”, but then that runs into the complication that any given theorem can have many different proofs, so it’s really more like “Is there any proof of this theorem that has structure similar to any proof of this other theorem?”. But to answer that requires familiarity with many different proofs of both. Or you could phrase it as “Does the proof of one of these follow logically from the other”, or perhaps both ways, but again, that’s not as well-defined as you’d think, because logically, any the proof of any theorem can be regarded as implied by any other theorem. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [August 11, 2025, 9:49pm UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/5 "2025-08-11T21:49:40Z")
Sure stumped me. There might be a theorem in projective geometry that covers both cases. A late colleague of mine would likely know the answer, but I know when I am out of my depth. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [August 11, 2025, 11:45pm UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/6 "2025-08-11T23:45:05Z")
> [@Chronos](#): > > “Are these things mathematically related?” is a difficult question to answer, because in some sense, all things in mathematics are related. I think one way to answer this is to ask given the three circles used to create the Monge line, is the intersection of the three Euler lines on that line? Or maybe does each Euler line intersect the Monge line at one of the points used to create it. Something like that. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [August 12, 2025, 12:07am UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/7 "2025-08-12T00:07:55Z")
Probably both theorems can be proved using Desargues’s Theorem or something like that— however, to me, “are the two lines mathematically related” seems to ask whether one problem can be transformed into an instance of the other, or whether both are special cases of a more general configuration. ---
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [August 12, 2025, 12:35pm UTC](https://boards.straightdope.com/t/monges-theorem-and-the-euler-line/1021707/8 "2025-08-12T12:35:24Z")
> [@DPRK](#): > > or whether both are special cases of a more general configuration. I was figuring _that_ would be the case.