Graphing stuff :/

thedarktiger · 2014-03-14 16:17:21

Let A = (1,2), B = (0,1), and C = (5,0). There exists a point Q and a constant k such that for any point P, PA^2 + PB^2 + PC^2 = 3PQ^2 + k. Find the point Q and the constant k. What is the significance of point Q with respect to triangle ABC?

thanks

Bob · 2014-03-16 01:43:47

hi

The most challenging yet.

I'm still a long way from a proper answer, but I thought it might be worth posting what I've found out so far.

I constructed the triangle and placed P and Q anywhere.

Then I computed an expression for k.

I moved Q to a new position; then tried out the expression as I moved P about.

I've gradually 'homed in' on a point that appears to satisfy the condition that k is constant (approximately).

It looks like it's the centroid (where the medians cross) and k is about 140.

So I'll take that as a working hypothesis and see if I can prove it.

Bob

Bob · 2014-03-16 05:04:51

I've been for a walk and worked out what to do:

First find the centroid. It's one third of the way up any median; use the median from (2.5,0) to (1,2) ; the centroid is at (2,1)

Let P be any point (x,y).

I shall work out k and hope to show it is independent of x and y.

So that proves the required result. Q is the centroid and k = 16. Why did I think it was 140 before. Well it looks like my software did not vary the measurements to allow for my re-scaling of the axes. I could work out the scaling factor but why bother? The result is found independent of my diagram.

Bob

note My software has called P the point D and Q the point F. E was a mistake.

Bob · 2014-03-16 05:05:39

Here's my diagram.

B

thedarktiger · 2014-03-17 00:37:00

Wow thanks! What software? Mathematica?
Wow these problems keep getting harder

Bob · 2014-03-17 01:01:48

When I saw your two geometry problems I thought the other one would be straight forward and this one tough. I've spent hours on the other one and got nowhere. Thank goodness Nehushtan came up with a way to do it!!

Rather in desperation I thought I'd try this one basically using trial and improvement to home in on a solution. My software is called Geometer's Sketchbook and, like Geogebra, it is a vector graphics program. As I got close to the 'correct' Q, it looked like Q might be at the centroid so I constructed it by intersecting the medians and moved Q on top. As P moved about, the change in k was very small so that's when I decided to make my first post. I was dragged out on a walk, but while walking around, I realised it wouldn't take much algebra to prove or disprove. When I got back, I tried it and it came out as shown.

The success to hours spent ratio is quite low compared with 'triangles', so my personal rating is that this one isn't so hard. Research to find out what Q is seems to be a major part of the question, so it's not really exam material is it? I have one more idea for 'triangles', which I'm about to try. What I'm looking for is a solution using angle sums, circles, parallels and such like.

Bob

Math Is Fun Forum

#1 2014-03-14 16:17:21

Graphing stuff :/

#2 2014-03-16 01:43:47

Re: Graphing stuff :/

#3 2014-03-16 05:04:51

Re: Graphing stuff :/

#4 2014-03-16 05:05:39

Re: Graphing stuff :/

#5 2014-03-17 00:37:00

Re: Graphing stuff :/

#6 2014-03-17 01:01:48

Re: Graphing stuff :/

Board footer