INMO - 2004

Geometry Level pending

Consider a convex quadrilateral A B C D ABCD , in which K , L , M , N K,L,M,N are the midpoints of A B , B C , C A , AB, BC, CA, & A D AD respectively. Suppose

B D BD bisects K M KM at Q Q ;

Q A = Q B = Q C = Q D QA = QB = QC = QD ;

L K L M = C D C B \frac{LK}{LM} = \frac{CD}{CB} ;

then, find the arithmetic mean of ∠ A , ∠ B \angle A, \angle B & ∠ C \angle C taken in degrees .


The answer is 90.

Ameya Salankar by Ameya Salankar , 23, India

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2 solutions

Michael Mendrin
Apr 16, 2014

The sum ∠A + ∠B + ∠C + ∠D for any quadrilateral is always 360? What do we need the rest of the details for?

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Well I edited that now, sorry for the mistake, sir.

Ameya Salankar - 7 years, 1 month ago
Joe Bobby
Jan 30, 2015

From the equality of diagonals and that they bisect each other, the figure is either a square or a rectangle. Since all the angels are 90˚ the average will obviously be 90˚.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

You can't say that it's a diagonal. Prove it Joe!

Kishore S. Shenoy - 5 years, 9 months ago

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