In a quadrilateral $\mathrm{ABCD}$, given that $\angle \mathrm{A}+\angle \mathrm{D}=90^{\circ}$. Prove that $\mathrm{AC}^{2}+\mathrm{BD}^{2}=\mathrm{AD}^{2}+\mathrm{BC}^{2}$.
Given: A quadrilateral $\mathrm{ABCD}$ where $\angle \mathrm{A}+\angle \mathrm{D}=90^{\circ}$.
To prove: $\mathrm{AC}^{2}+\mathrm{BD}^{2}=\mathrm{AD}^{2}+\mathrm{BC}^{2}$
Construction: Extend AB and CD to intersect at O.
Proof:
In $\triangle \mathrm{AOD}, \angle \mathrm{A}+\angle \mathrm{O}+\angle \mathrm{D}=180^{\circ}$
$\Rightarrow \angle O=90^{\circ}\left[\angle A+\angle D=90^{\circ}\right]$
Apply Pythagoras Theorem in $\triangle A O C$ and $\triangle B O D$,
$\mathrm{AC}^{2}=\mathrm{AO}^{2}+\mathrm{OC}^{2}$
$\mathrm{BD}^{2}=\mathrm{OB}^{2}+\mathrm{OD}^{2}$
$\therefore \mathrm{AC}^{2}+\mathrm{BD}^{2}=\left(\mathrm{AO}^{2}+\mathrm{OD}^{2}\right)+\left(\mathrm{OC}^{2}+\mathrm{OB}^{2}\right)$
$\Rightarrow \mathrm{AC}^{2}+\mathrm{BD}^{2}=\mathrm{AD}^{2}+\mathrm{BC}^{2}$
This proves the given relation.
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