Lemma XIX.
*Invenire punctum*P*, a quo si rectæ quatuor*PQ*,*PR*,*PS*,*PT*ad alias totidem positione datas rectas*AB*,*CD*,*AC*,*BD*singulæ ad singulas in datis angulis ducantur, rectangulum sub duabus ductis,*PQ*×*PR*, sit ad rectangulum sub aliis duabus,*PS*×*PT*, in data ratione.*
Lineæ AB, CD, ad quas rectæ duæ PQ, PR, unum rectangulorum continentes ducuntur, conveniant cum aliis duabus positione datis lineis in punctis A, B, C, D. Ab eorum aliquo A age rectam quamlibet AH, in qua velis punctum P reperiri. Secet ea lineas oppositas BD, CD, nimirum BD in H & CD in I, & ob datos omnes angulos figuræ, dabuntur rationes PQ ad PA & PA ad PS, adeoq; ratio PQ ad PS. Auferendo hanc a data ratione PQ × PR ad PS × PT, dabitur ratio PR ad PT, & addendo datas rationes PI ad PR, & PT ad PH dabitur ratio PI ad PH atq; adeo punctum P. Q. E. I.