ABCD a square and M a point on or inside the square.Where to place point M so that MA+MB is minimum

ABCD a square and M a point on or inside the square.Where to place point M so that MA+MA is minimum
olympiad question. Very classic geometric problem.Elegant math problem. #maths #calculus #olympiad
This is a minimization problem. We are trying to find a point. M, such that distance from a Plus, the distance from B to M is minimum. We are inside a square. So we now how to find the point That's close to A and B at the same time. It's the center of the square But here this point am is changing. So we want to find ampsads that M A plus MB is minimum How can we do that? Okay, if we fix a And B. And we Identically have m equal a and B or B. Then we see that M A plus m. B is just a b. And in this case, the distance is minimal On the other half. If M is between A and B, we see that M A plus MB is equal to a b, Okay? If m is on any of the other side, see or D, then We see that M A plus M Is the same as MB plus MD And this is larger than Distance that we need On the other half. If we try to use the triangular inequality for the triangle, We see that in the triangle, a b, we have a B is less than M A plus MB. So we see that this distance is at least an M A plus and B is at least equal to a B1. M is either a or b or m is between A and B. So that's what we're gonna try to prove To prove his result. We're going to use some geometric problems. Using a ruler and a campus. So that campus, not campus Campus. Campus Compass Compass. Okay, that's what we need. We're going to use a ruler and a campus And campus. Not compass Campus. Compass Compass. Okay, so that's what we're gonna use here. This is a very famous problem. Later, we're gonna see if we can find another point here, When we try to find the minimum distance, M, A plus MB, plus MC inside Square. What should we place? What should we place our point? And so that distance M A plus m, b plus MC is minimal. This is a nice old geometric problem that we try to solve using a ruler and a campus Campus Compass. Compass compass, Okay? And the key here is that we will use geometry. We are not going to use analytical geometry because that makes the problem, very easy. So what do you think? What what what's, what's the solution? What should we place? The point M? So that the distance M, A plus m b is minimal. Thank you.

[@ViliamF.]

The title and the description have a typo. "MA+MA" -> "MA+MB"
And the answer doesn't really make sense. The sum |MA|+|MB| has the same minimal value (the side length of the square, which is the length of the line segment AB) when M is any point on that side AB, including A and B themselves.

[@Archimedes_Notes]

@@ViliamF. What is tbis typo? I see now . Yes. The MA+MB THANK YOU. I DID FIX IT.

[@Archimedes_Notes]

@@ViliamF. I did not give the complete solution yet. That is a discussion of the problem. The answer will be on the other video concerning Fermat problem