Thursday, January 4, 2018

Geometry Problem 1352: Rectangles, Sum of Squared Distances

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

Geometry Problem 1352: Rectangles, Sum of Squared Distances.



    Let 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

  2. Draw PP1 perpend to BC extended, PP2 perpend to AD extended
    b²=(BC+CP1)²+PP1² , d²=DP2²+PP2²
    c²=PP1²+CP1² , a²=(AD+DP2)²+PP2²
    => b²+d²=a²+c²

  3. Very straightforward using Appollonius.

    a^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
    Sri Lanka