143416VERA CIRCULI
Ducatur recta D L ſegmentum tangens in puncto I, &
rectis B F, F P, occurrens in punctis D, L, ita ut com-
pleatur polygonum A B D L P.
rectis B F, F P, occurrens in punctis D, L, ita ut com-
pleatur polygonum A B D L P.
PROP. II. THEOREMA.
Dico trapezia A B F P, A B I P ſimul, eſſe ad du-
plum trapezii A B I P, ſicut trapezium A B F P ad polygonum A B D L P.
11TAB. XLIII. plum trapezii A B I P, ſicut trapezium A B F P ad polygonum A B D L P.
Fig. 1. 2. 3.
Quoniam recta A F, ducta per contactum rectæ D L cum
ſegmento, ducitur etiam per concurſum duarum recta-
rum F B, F P, rectam D L terminantium & ſegmen-
tum in duobus punctis tangentium; igitur recta D L bifa-
riam ſecatur in puncto I; & proinde triangulum F D I æ-
quale eſt triangulo F I L, at triangulum A B F æquale eſt
triangulo A P F; & igitur trapezium A B D I æquale eſt
trapezio A P L I; trapezium ergo A P L I dimidium eſt
polygoni A B D L P. ducatur recta A L: manifeſtum eſt
ex præcedentis demonſtratione triangulum A I L eſſe æqua-
le triangulo A L P; ſed ut triangulum A L F ad triangu-
lum A L I ita F A ad A I, & ut F A ad A I ita trapezium
A B F P ad trapezium A B I P; & igitur ut trapezium
A B F P ad trapezium A B I P; ita triangulum A L F ad
triangulum A L I; & componendo, ut trapezia A B F P,
A B I P ſimul, ad trapezium A B I P, ita triangulum A F L
& triangulum A I L ſimul, hoc eſt triangulum A F P, ad
triangulum A I L: & conſequentes duplicando, ut trape-
zia A B F P, A B I P ſimul, ad duplum trapezii A B I P,
ita triangulum A F P, ad trapezium A I L P: at triangu-
lum A F P eſt dimidium trapezii A B F P, & trapezium
A I L P eſt dimidium polygoni A B D L P; & igitur ut
trapezia A B F P, A B I P ſimul, ad duplum trapezii
A B I P, ita trapezium A B F P ad polygonum A B D L P,
quod demonſtrare oportuit.
ſegmento, ducitur etiam per concurſum duarum recta-
rum F B, F P, rectam D L terminantium & ſegmen-
tum in duobus punctis tangentium; igitur recta D L bifa-
riam ſecatur in puncto I; & proinde triangulum F D I æ-
quale eſt triangulo F I L, at triangulum A B F æquale eſt
triangulo A P F; & igitur trapezium A B D I æquale eſt
trapezio A P L I; trapezium ergo A P L I dimidium eſt
polygoni A B D L P. ducatur recta A L: manifeſtum eſt
ex præcedentis demonſtratione triangulum A I L eſſe æqua-
le triangulo A L P; ſed ut triangulum A L F ad triangu-
lum A L I ita F A ad A I, & ut F A ad A I ita trapezium
A B F P ad trapezium A B I P; & igitur ut trapezium
A B F P ad trapezium A B I P; ita triangulum A L F ad
triangulum A L I; & componendo, ut trapezia A B F P,
A B I P ſimul, ad trapezium A B I P, ita triangulum A F L
& triangulum A I L ſimul, hoc eſt triangulum A F P, ad
triangulum A I L: & conſequentes duplicando, ut trape-
zia A B F P, A B I P ſimul, ad duplum trapezii A B I P,
ita triangulum A F P, ad trapezium A I L P: at triangu-
lum A F P eſt dimidium trapezii A B F P, & trapezium
A I L P eſt dimidium polygoni A B D L P; & igitur ut
trapezia A B F P, A B I P ſimul, ad duplum trapezii
A B I P, ita trapezium A B F P ad polygonum A B D L P,
quod demonſtrare oportuit.
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