Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.
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Thursday, January 4, 2018
Geometry Problem 1352: Rectangles, Sum of Squared Distances
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https://photos.app.goo.gl/tL43BsQs4gmUj4YL2
ReplyDeleteLet AC meet BD at M and FH meet EG at N
M and N are the midpoint of diagonals of rectangles ABCD and EFGH
In triangle BPD we have b^2+d^2= 2. PM^2+BD^2/2
Similarly in triangle APC we have a^2+c^2= 2.PM^2+ AC^2/2
Since BD= AC => b^2+d^2= a^2+c^2
Similarly with rectangle EFGH we also have f^2+h^2= e^2+g^2
Add above 2 expressions side by side we will get the result
Draw PP1 perpend to BC extended, PP2 perpend to AD extended
ReplyDeleteb²=(BC+CP1)²+PP1² , d²=DP2²+PP2²
c²=PP1²+CP1² , a²=(AD+DP2)²+PP2²
=> b²+d²=a²+c²
Very straightforward using Appollonius.
ReplyDeletea^2 + c^2 = b^2 + d^2 = 2.PO^2 + BD^2/2 and similarly for EFGH and the result follows upon addition
Sumith Peiris
Moratuwa
Sri Lanka