Translation examples
Nyt Pythagoraan lauseen mukaan, A P 2 = A w 2 + w P 2 = A w 2 + A z 2 {\displaystyle AP^{2}=Aw^{2}+wP^{2}=Aw^{2}+Az^{2}} Näin voimme laskea pisteen P ja suorakulmion kolmen muun kulman välisten etäisyyksien neliöt: P C 2 = w B 2 + z D 2 , {\displaystyle PC^{2}=wB^{2}+zD^{2},} B P 2 = w B 2 + A z 2 , {\displaystyle BP^{2}=wB^{2}+Az^{2},} ja P D 2 = z D 2 + A w 2 . {\displaystyle PD^{2}=zD^{2}+Aw^{2}.} Täten: A P 2 + P C 2 = ( A w 2 + A z 2 ) + ( w B 2 + z D 2 ) = ( w B 2 + A z 2 ) + ( z D 2 + A w 2 ) = B P 2 + P D 2 . {\displaystyle AP^{2}+PC^{2}=(Aw^{2}+Az^{2})+(wB^{2}+zD^{2})=(wB^{2}+Az^{2})+(zD^{2}+Aw^{2})=BP^{2}+PD^{2}.\,} Lauseen nimi tulee siitä, kun piirretään pisteestä P viivat suorakulmion kulmiin sekä todistuksessa käytettävät suorakulmion sivujen kanssa kohtisuorat viivat, lopputulos muistuttaa joidenkin mielestä Yhdistyneen kuningaskunnan lippua.
By applying the Pythagorean theorem to the right triangle AWP, and observing that WP = AZ, it follows that A P 2 = A W 2 + W P 2 = A W 2 + A Z 2 {\displaystyle AP^{2}=AW^{2}+WP^{2}=AW^{2}+AZ^{2}} and by a similar argument the squares of the lengths of the distances from P to the other three corners can be calculated as P C 2 = W B 2 + Z D 2 , {\displaystyle PC^{2}=WB^{2}+ZD^{2},} B P 2 = W B 2 + A Z 2 , {\displaystyle BP^{2}=WB^{2}+AZ^{2},} and P D 2 = Z D 2 + A W 2 . {\displaystyle PD^{2}=ZD^{2}+AW^{2}.} Therefore: A P 2 + P C 2 = ( A W 2 + A Z 2 ) + ( W B 2 + Z D 2 ) = ( W B 2 + A Z 2 ) + ( Z D 2 + A W 2 ) = B P 2 + P D 2 {\displaystyle {\begin{aligned}AP^{2}+PC^{2}&=\left(AW^{2}+AZ^{2}\right)+\left(WB^{2}+ZD^{2}\right)\\&=\left(WB^{2}+AZ^{2}\right)+\left(ZD^{2}+AW^{2}\right)\\&=BP^{2}+PD^{2}\end{aligned}}} This theorem takes its name from the fact that, when the line segments from P to the corners of the rectangle are drawn, together with the perpendicular lines used in the proof, the completed figure somewhat resembles a Union Flag.
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