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Difficulty: HardProperties of Quadrilaterals

For any convex quadrilateral ABCDABCD, is the statement that AB2+CD2=BC2+DA2AB^2 + CD^2 = BC^2 + DA^2 if and only if the diagonals ACAC and BDBD are perpendicular true or false?

Answer: Answer

Answer

The statement is true because the sum of the squares of the lengths of opposite sides in a convex quadrilateral is equal if and only if its diagonals intersect at right angles.
The statement is true because the equality of the sums of the squares of opposite sides is mathematically equivalent to the diagonals being perpendicular in any convex quadrilateral.

Step-by-Step Solution

1
Define the intersection point of the diagonals ACAC and BDBD as PP and the angle of intersection as θ\theta.
Four triangles are formed: △APB\triangle APB, △BPC\triangle BPC, △CPD\triangle CPD, and △DPA\triangle DPA, with angles at PP being θ\theta and 180∘−θ180^\circ - \theta.
This establishes a geometric frame of reference to relate side lengths to diagonal segments.
2
Apply the Law of Cosines to express the square of each side length in terms of the diagonal segments APAP, BPBP, CPCP, and DPDP.
AB2=AP2+BP2−2(AP)(BP)cos⁡θAB^2 = AP^2 + BP^2 - 2(AP)(BP)\cos\theta, CD2=CP2+DP2−2(CP)(DP)cos⁡θCD^2 = CP^2 + DP^2 - 2(CP)(DP)\cos\theta, BC2=BP2+CP2+2(BP)(CP)cos⁡θBC^2 = BP^2 + CP^2 + 2(BP)(CP)\cos\theta, and DA2=DP2+AP2+2(DP)(AP)cos⁡θDA^2 = DP^2 + AP^2 + 2(DP)(AP)\cos\theta.
This links the boundary side lengths of the quadrilateral to its internal diagonals.
3
Sum the squares of the opposite sides and compute their difference: (AB2+CD2)−(BC2+DA2)(AB^2 + CD^2) - (BC^2 + DA^2).
(AB2+CD2)−(BC2+DA2)=−2(AP⋅BP+CP⋅DP+BP⋅CP+DP⋅AP)cos⁡θ(AB^2 + CD^2) - (BC^2 + DA^2) = -2(AP\cdot BP + CP\cdot DP + BP\cdot CP + DP\cdot AP)\cos\theta.
This algebraic combination isolates the term involving the angle of intersection.
4
Factor the coefficient of −2cos⁡θ-2\cos\theta and simplify the relation.
(AB2+CD2)−(BC2+DA2)=−2(AP+CP)(BP+DP)cos⁡θ=−2(AC)(BD)cos⁡θ(AB^2 + CD^2) - (BC^2 + DA^2) = -2(AP + CP)(BP + DP)\cos\theta = -2(AC)(BD)\cos\theta.
Factoring groups the individual segments into the full lengths of the diagonals ACAC and BDBD.
5
Analyze the condition for the difference to be zero.
AB2+CD2=BC2+DA2  ⟺  cos⁡θ=0  ⟺  θ=90∘AB^2 + CD^2 = BC^2 + DA^2 \iff \cos\theta = 0 \iff \theta = 90^\circ.
Since the lengths ACAC and BDBD must be positive, the difference is zero if and only if the diagonals are perpendicular.

Key Concept

Orthodiagonal quadrilateral properties and diagonal relations
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Topics

Properties of Quadrilaterals

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