Points K, L, M and N are marked in the convex quadrilateral ABCD – the midpoints of the sides

Points K, L, M and N are marked in the convex quadrilateral ABCD – the midpoints of the sides AD, AB, BC, and CD, respectively. Find the ratio of the area of the quadrilateral ABCD to the area of the quadrilateral KLMN.

For triangles ABC and ACD, segments KL and IM are centerlines that are parallel to AC.
For triangles ABD and BCD, segments KI and LM are midlines that are parallel to BD. The KLMN quadrilateral is a parallelogram. The middle lines bisect the AJ and CG heights.

Diagonal BD is divided by the middle lines of the triangles:

BF = FO OH = HD;

BF + HD = FO + OH;

BD = BF + FO + OH + HD;

BD = 2 (FO + OH) = 2 * FH;

FH = BD / 2;

PJ = AJ / 2, where AJ is the height of triangle ABD.

SKFHI = FH * PJ = BD / 2 * AJ / 2 = (BD * AJ) / 4.

SABD = (BD * AJ) / 2.

SKFHI = SABD / 2.

Similarly, one can prove that SFLMH = SBCD / 2.

SKFHI + SFLMH = SABD / 2 + SBCD / 2 = (SABD + SBCD) / 2

SKLMN = SABCD / 2;

Answer: SABCD / SKLMN = 2.