20060308, 21:57  #1 
May 2004
New York City
5×7×11^{2} Posts 
Geometry Puzzle #2
Within a 3x3 square ABCD (corners) there is a point P
whose distances from A,B,C, and D respectively are sqrt2, sqrt5, sqrt8, and x. Find the value of x. 
20060309, 16:28  #2 
Bronze Medalist
Jan 2004
Mumbai,India
804_{16} Posts 
Geometry Puzzle
Excellent problem davar
Ans: sq. rt. 5 Hint : Dont try Heron's formula. Mally Last fiddled with by mfgoode on 20060309 at 16:30 
20060311, 00:16  #3 
Jan 2005
3 Posts 
AP+PC = sqrt2+sqrt8 = sqrt2+2sqrt2 = 3sqrt2
AC = 3sqrt2 =>P lies on AC. From the congruent triangles APB,APD we get PD=PB=sqrt5 
20060314, 15:59  #4 
Dec 2005
2^{2}×7^{2} Posts 
A little trick
Playing with Pythagoras: Let P be the given point, PX its orthogonal projection on AB and PY its orthogonal projection on AD. Let x=d(A,PX), y=d(A,PY).
Then we have x^2+y^2 = 2 ( I) x^2+(3y)^2 = 5 ( II) (3x)^2 + (3y)^2 = 8 (III) and we want to find (3x)^2+ y^2 (IV) We have the following formula: III=IV+III which leads to IV=IIIII+I = 85+2 =5. Taking square root gives the result sorry for the format, do not know how to blacken text so wrote in white 
20060314, 16:48  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Geometry Puzzle
Quote:
Well lets say you are not as green as you're cabbage looking! Mally 

20060314, 16:54  #6 
Dec 2005
C4_{16} Posts 
More generally speaking
we have the following result:
PA^2+PD^2=PB^2+PC^2 which follows directly from the given formula, but I suppose there might be a geometrical argument 
20060314, 17:03  #7 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Geometry Puzzle
Just draw the figure roughly. The answer pops out by mere inspection!
Mally 
20060314, 17:33  #8 
Dec 2005
196_{10} Posts 
The general figure
I am not seeing it, I place a point somewhere in a rectangle and then I can just see by drawing the lines from the vertices to this point that this solves it all ?
Drawing lines parallel to AB and AD through P helps seeing the solution, but if that is considered too long... 
20060315, 16:13  #9  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}×3^{3}×19 Posts 
Geometry Puzzle#2
Quote:
Seriously Kees you are quite right. // lines do help. Maybe the problem is with your vision using three 'seeings' in one para ! Perhaps you used the invisible colour and could not 'see' the problem at all? Just joking Kees. How about a hint to your number? Mally 

20060315, 22:27  #10 
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Kees,
Put " [ spoiler ] " (without the spaces!) before the text you wish to blackout, and " [ / spoiler ] " (without the spaces!) after that text. Then, those of us using offwhite backgrounds will not unwillingly see your text. :) (P.S., nice avatar!) 
20060316, 08:41  #11 
Dec 2005
C4_{16} Posts 
Spoler test
so this should be hidden thanks for the tip 
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